Sucesión (matemáticas) - Wikipedia, a enciclopedia libre

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="gl" dir="ltr"> <head> <meta charset="UTF-8"> <title>Sucesión (matemáticas) - Wikipedia, a enciclopedia libre</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )glwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t.",".\t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","xaneiro","febreiro","marzo","abril","maio","xuño","xullo","agosto","setembro","outubro","novembro","decembro"],"wgRequestId":"e1bf080b-977c-4071-a5d5-abf9ef0f4a46","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Sucesión_(matemáticas)","wgTitle":"Sucesión (matemáticas)","wgCurRevisionId":6863764,"wgRevisionId":6863764,"wgArticleId":2323,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Wikiproxecto 1000 artigos de calidade para alumnado de 12 a 16 anos/Matemáticas","Wikipedia:Páxinas con traducións non revisadas","Matemáticas"],"wgPageViewLanguage":"gl","wgPageContentLanguage":"gl","wgPageContentModel":"wikitext","wgRelevantPageName":"Sucesión_(matemáticas)","wgRelevantArticleId":2323,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom": "Sucesión_matemática","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"gl","pageLanguageDir":"ltr","pageVariantFallbacks":"gl"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgInternalRedirectTargetUrl":"/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)","wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q133250","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.charinsert-styles":"ready","ext.gadget.PortalClass":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.refToolbar","ext.gadget.charinsert","ext.urlShortener.toolbar", "ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=gl&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=gl&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=gl&modules=ext.gadget.PortalClass%2Ccharinsert-styles&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=gl&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/1200px-Cauchy_sequence_illustration2.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="669"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/800px-Cauchy_sequence_illustration2.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="446"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/640px-Cauchy_sequence_illustration2.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="357"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Sucesión (matemáticas) - Wikipedia, a enciclopedia libre"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//gl.m.wikipedia.org/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)"> <link rel="alternate" type="application/x-wiki" title="Editar" href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (gl)"> <link rel="EditURI" type="application/rsd+xml" href="//gl.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://gl.wikipedia.org/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.gl"> <link rel="alternate" type="application/atom+xml" title="Fonte Atom de novas de Wikipedia" href="/w/index.php?title=Especial:Cambios_recentes&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Sucesión_matemáticas rootpage-Sucesión_matemáticas skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Saltar ao contido</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Menú principal" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Menú principal</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Menú principal</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">agochar</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navegación </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Portada" title="Visitar a páxina principal [z]" accesskey="z"><span>Portada</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Portal_da_comunidade" title="Información acerca do proxecto, do que pode facer e dos lugares onde atopar as cousas"><span>Portal da comunidade</span></a></li><li id="n-A-Taberna" class="mw-list-item"><a href="/wiki/Wikipedia:A_Taberna"><span>A Taberna</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Wikipedia:Actualidade" title="Información acerca de acontecementos de actualidade"><span>Actualidade</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Especial:Cambios_recentes" title="A lista de modificacións recentes no wiki [r]" accesskey="r"><span>Cambios recentes</span></a></li><li id="n-Artigos-de-calidade" class="mw-list-item"><a href="/wiki/Wikipedia:Artigos_de_calidade"><span>Artigos de calidade</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Especial:Ao_chou" title="Cargar unha páxina ao chou [x]" accesskey="x"><span>Páxina ao chou</span></a></li><li id="n-help" class="mw-list-item"><a href="/wiki/Wikipedia:Axuda" title="O lugar para informarse"><span>Axuda</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Portada" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="a Wikipedia en galego" src="/static/images/mobile/copyright/wikipedia-tagline-gl.svg" width="118" height="13" style="width: 7.375em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Especial:Procurar" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Procurar neste wiki [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Procura</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Procurar en Wikipedia" aria-label="Procurar en Wikipedia" autocapitalize="sentences" title="Procurar neste wiki [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Especial:Procurar"> </div> <button class="cdx-button cdx-search-input__end-button">Procurar</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Ferramentas persoais"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Aparencia"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Cambia a aparencia do tamaño da fonte, o ancho e a cor da páxina" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Aparencia" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Aparencia</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=gl.wikipedia.org&uselang=gl" class=""><span>Doazóns</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Crear_unha_conta&returnto=Sucesi%C3%B3n+%28matem%C3%A1ticas%29" title="É recomendable que cree unha conta e acceda ao sistema, se ben non é obrigatorio" class=""><span>Crear unha conta</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Especial:Iniciar_sesi%C3%B3n&returnto=Sucesi%C3%B3n+%28matem%C3%A1ticas%29" title="É recomendable que se rexistre, se ben non é obrigatorio [o]" accesskey="o" class=""><span>Acceder ao sistema</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Máis opcións" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas persoais" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Ferramentas persoais</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Menú de usuario" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=gl.wikipedia.org&uselang=gl"><span>Doazóns</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Crear_unha_conta&returnto=Sucesi%C3%B3n+%28matem%C3%A1ticas%29" title="É recomendable que cree unha conta e acceda ao sistema, se ben non é obrigatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Crear unha conta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Iniciar_sesi%C3%B3n&returnto=Sucesi%C3%B3n+%28matem%C3%A1ticas%29" title="É recomendable que se rexistre, se ben non é obrigatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Acceder ao sistema</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Páxinas para os editores sen a sesión iniciada <a href="/wiki/Axuda:Introduci%C3%B3n" aria-label="Máis información sobre a edición"><span>máis información</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Especial:As_mi%C3%B1as_contribuci%C3%B3ns" title="Unha lista das modificacións feitas desde este enderezo IP [y]" accesskey="y"><span>Contribucións</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:A_mi%C3%B1a_conversa" title="Conversa acerca de edicións feitas desde este enderezo IP [n]" accesskey="n"><span>Conversa</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contidos" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">agochar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inicio</div> </a> </li> <li id="toc-Notación_e_exemplos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notación_e_exemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Notación e exemplos</span> </div> </a> <button aria-controls="toc-Notación_e_exemplos-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Notación e exemplos"</span> </button> <ul id="toc-Notación_e_exemplos-sublist" class="vector-toc-list"> <li id="toc-Exemplos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Exemplos</span> </div> </a> <ul id="toc-Exemplos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Indexación" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Indexación"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Indexación</span> </div> </a> <ul id="toc-Indexación-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definición_dunha_secuencia_por_recursividade" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición_dunha_secuencia_por_recursividade"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Definición dunha secuencia por recursividade</span> </div> </a> <ul id="toc-Definición_dunha_secuencia_por_recursividade-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definición_formal_e_propiedades_básicas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definición_formal_e_propiedades_básicas"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definición formal e propiedades básicas</span> </div> </a> <button aria-controls="toc-Definición_formal_e_propiedades_básicas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Definición formal e propiedades básicas"</span> </button> <ul id="toc-Definición_formal_e_propiedades_básicas-sublist" class="vector-toc-list"> <li id="toc-Definición" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Definición</span> </div> </a> <ul id="toc-Definición-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finita_e_infinita" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finita_e_infinita"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Finita e infinita</span> </div> </a> <ul id="toc-Finita_e_infinita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Crecente_e_decrecente" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Crecente_e_decrecente"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Crecente e decrecente</span> </div> </a> <ul id="toc-Crecente_e_decrecente-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitada" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitada"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Limitada</span> </div> </a> <ul id="toc-Limitada-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subsecuencias" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subsecuencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Subsecuencias</span> </div> </a> <ul id="toc-Subsecuencias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Outros_tipos_de_secuencias" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_tipos_de_secuencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Outros tipos de secuencias</span> </div> </a> <ul id="toc-Outros_tipos_de_secuencias-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Límites_e_converxencia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Límites_e_converxencia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Límites e converxencia</span> </div> </a> <button aria-controls="toc-Límites_e_converxencia-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Límites e converxencia"</span> </button> <ul id="toc-Límites_e_converxencia-sublist" class="vector-toc-list"> <li id="toc-Definición_formal_de_converxencia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definición_formal_de_converxencia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Definición formal de converxencia</span> </div> </a> <ul id="toc-Definición_formal_de_converxencia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aplicacións_e_resultados_importantes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aplicacións_e_resultados_importantes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Aplicacións e resultados importantes</span> </div> </a> <ul id="toc-Aplicacións_e_resultados_importantes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Secuencias_de_Cauchy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Secuencias_de_Cauchy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Secuencias de Cauchy</span> </div> </a> <ul id="toc-Secuencias_de_Cauchy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Límites_infinitos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Límites_infinitos"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Límites infinitos</span> </div> </a> <ul id="toc-Límites_infinitos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Serie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Serie"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Serie</span> </div> </a> <ul id="toc-Serie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Notas</span> </div> </a> <ul id="toc-Notas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Outros_artigos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outros_artigos"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Outros artigos</span> </div> </a> <ul id="toc-Outros_artigos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ligazóns_externas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligazóns_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Ligazóns externas</span> </div> </a> <ul id="toc-Ligazóns_externas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Mostrar ou agochar a táboa de contidos" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Mostrar ou agochar a táboa de contidos</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Sucesión (matemáticas)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artigo noutra lingua. Dispoñible en 75 linguas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-75" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">75 linguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%AA%D8%A7%D9%84%D9%8A%D8%A9" title="متتالية – árabe" lang="ar" hreflang="ar" data-title="متتالية" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Socesi%C3%B3n_matem%C3%A1tica" title="Socesión matemática – asturiano" lang="ast" hreflang="ast" data-title="Socesión matemática" data-language-autonym="Asturianu" data-language-local-name="asturiano" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Ard%C4%B1c%C4%B1ll%C4%B1q" title="Ardıcıllıq – acerbaixano" lang="az" hreflang="az" data-title="Ardıcıllıq" data-language-autonym="Azərbaycanca" data-language-local-name="acerbaixano" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AD%D2%99%D0%BC%D3%99-%D1%8D%D2%99%D0%BB%D0%B5%D0%BB%D0%B5%D0%BA" title="Эҙмә-эҙлелек – baxkir" lang="ba" hreflang="ba" data-title="Эҙмә-эҙлелек" data-language-autonym="Башҡортса" data-language-local-name="baxkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B4%D0%B8%D1%86%D0%B0" title="Редица – búlgaro" lang="bg" hreflang="bg" data-title="Редица" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%85%E0%A6%A8%E0%A7%81%E0%A6%95%E0%A7%8D%E0%A6%B0%E0%A6%AE" title="অনুক্রম – bengalí" lang="bn" hreflang="bn" data-title="অনুক্রম" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Niz" title="Niz – bosníaco" lang="bs" hreflang="bs" data-title="Niz" data-language-autonym="Bosanski" data-language-local-name="bosníaco" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Successi%C3%B3_(matem%C3%A0tiques)" title="Successió (matemàtiques) – catalán" lang="ca" hreflang="ca" data-title="Successió (matemàtiques)" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%D8%A7%D8%B4%DB%8C%DB%95%DA%A9%DB%8C" title="پاشیەکی – kurdo central" lang="ckb" hreflang="ckb" data-title="پاشیەکی" data-language-autonym="کوردی" data-language-local-name="kurdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Posloupnost" title="Posloupnost – checo" lang="cs" hreflang="cs" data-title="Posloupnost" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%BC%D0%BB%C4%83%D0%BD-%D1%85%D1%8B%C3%A7%D0%BB%C4%83%D0%BD%D0%BB%C4%83%D1%85" title="Умлăн-хыçлăнлăх – chuvaxo" lang="cv" hreflang="cv" data-title="Умлăн-хыçлăнлăх" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Talf%C3%B8lge" title="Talfølge – dinamarqués" lang="da" hreflang="da" data-title="Talfølge" data-language-autonym="Dansk" data-language-local-name="dinamarqués" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Folge_(Mathematik)" title="Folge (Mathematik) – alemán" lang="de" hreflang="de" data-title="Folge (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BA%CE%BF%CE%BB%CE%BF%CF%85%CE%B8%CE%AF%CE%B1" title="Ακολουθία – grego" lang="el" hreflang="el" data-title="Ακολουθία" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Sequence" title="Sequence – inglés" lang="en" hreflang="en" data-title="Sequence" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Vico" title="Vico – esperanto" lang="eo" hreflang="eo" data-title="Vico" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Sucesi%C3%B3n_(matem%C3%A1tica)" title="Sucesión (matemática) – español" lang="es" hreflang="es" data-title="Sucesión (matemática)" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Jada" title="Jada – estoniano" lang="et" hreflang="et" data-title="Jada" data-language-autonym="Eesti" data-language-local-name="estoniano" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Segida" title="Segida – éuscaro" lang="eu" hreflang="eu" data-title="Segida" data-language-autonym="Euskara" data-language-local-name="éuscaro" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%86%D8%A8%D8%A7%D9%84%D9%87" title="دنباله – persa" lang="fa" hreflang="fa" data-title="دنباله" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Lukujono" title="Lukujono – finés" lang="fi" hreflang="fi" data-title="Lukujono" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Suite_(math%C3%A9matiques)" title="Suite (mathématiques) – francés" lang="fr" hreflang="fr" data-title="Suite (mathématiques)" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Seicheamh" title="Seicheamh – irlandés" lang="ga" hreflang="ga" data-title="Seicheamh" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה) – hebreo" lang="he" hreflang="he" data-title="סדרה (מתמטיקה)" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="अनुक्रम – hindi" lang="hi" hreflang="hi" data-title="अनुक्रम" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Niz" title="Niz – croata" lang="hr" hreflang="hr" data-title="Niz" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Sorozat_(matematika)" title="Sorozat (matematika) – húngaro" lang="hu" hreflang="hu" data-title="Sorozat (matematika)" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%BB%D5%B8%D6%80%D5%A4%D5%A1%D5%AF%D5%A1%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Հաջորդականություն (մաթեմատիկա) – armenio" lang="hy" hreflang="hy" data-title="Հաջորդականություն (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="armenio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Sequentia_(mathematica)" title="Sequentia (mathematica) – interlingua" lang="ia" hreflang="ia" data-title="Sequentia (mathematica)" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Barisan" title="Barisan – indonesio" lang="id" hreflang="id" data-title="Barisan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Sequo" title="Sequo – ido" lang="io" hreflang="io" data-title="Sequo" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Runa" title="Runa – islandés" lang="is" hreflang="is" data-title="Runa" data-language-autonym="Íslenska" data-language-local-name="islandés" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Successione_(matematica)" title="Successione (matematica) – italiano" lang="it" hreflang="it" data-title="Successione (matematica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%97_(%E6%95%B0%E5%AD%A6)" title="列 (数学) – xaponés" lang="ja" hreflang="ja" data-title="列 (数学)" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%98%E1%83%9B%E1%83%93%E1%83%94%E1%83%95%E1%83%A0%E1%83%9D%E1%83%91%E1%83%90" title="მიმდევრობა – xeorxiano" lang="ka" hreflang="ka" data-title="მიმდევრობა" data-language-autonym="ქართული" data-language-local-name="xeorxiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%96%D0%B7%D0%B1%D0%B5%D0%BA_(%D0%B6%D0%B8%D1%8B%D0%BD)" title="Тізбек (жиын) – kazako" lang="kk" hreflang="kk" data-title="Тізбек (жиын)" data-language-autonym="Қазақша" data-language-local-name="kazako" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%EC%97%B4" title="수열 – coreano" lang="ko" hreflang="ko" data-title="수열" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Sequentia_(mathematica)" title="Sequentia (mathematica) – latín" lang="la" hreflang="la" data-title="Sequentia (mathematica)" data-language-autonym="Latina" data-language-local-name="latín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Suite_(Mathematik)" title="Suite (Mathematik) – luxemburgués" lang="lb" hreflang="lb" data-title="Suite (Mathematik)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburgués" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Succession_(matematega)" title="Succession (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Succession (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Seka" title="Seka – lituano" lang="lt" hreflang="lt" data-title="Seka" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Virkne" title="Virkne – letón" lang="lv" hreflang="lv" data-title="Virkne" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B8%D0%B7%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Низа (математика) – macedonio" lang="mk" hreflang="mk" data-title="Низа (математика)" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B5%81%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%E0%B4%82" title="അനുക്രമം – malabar" lang="ml" hreflang="ml" data-title="അനുക്രമം" data-language-autonym="മലയാളം" data-language-local-name="malabar" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Jujukan" title="Jujukan – malaio" lang="ms" hreflang="ms" data-title="Jujukan" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mwl mw-list-item"><a href="https://mwl.wikipedia.org/wiki/Sequ%C3%A9ncia_(matem%C3%A1tica)" title="Sequéncia (matemática) – mirandés" lang="mwl" hreflang="mwl" data-title="Sequéncia (matemática)" data-language-autonym="Mirandés" data-language-local-name="mirandés" class="interlanguage-link-target"><span>Mirandés</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%85%E0%A4%A8%E0%A5%81%E0%A4%95%E0%A5%8D%E0%A4%B0%E0%A4%AE" title="अनुक्रम – nepalí" lang="ne" hreflang="ne" data-title="अनुक्रम" data-language-autonym="नेपाली" data-language-local-name="nepalí" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Rij_(wiskunde)" title="Rij (wiskunde) – neerlandés" lang="nl" hreflang="nl" data-title="Rij (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/F%C3%B8lgje" title="Følgje – noruegués nynorsk" lang="nn" hreflang="nn" data-title="Følgje" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegués nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/F%C3%B8lge_(matematikk)" title="Følge (matematikk) – noruegués bokmål" lang="nb" hreflang="nb" data-title="Følge (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="noruegués bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ci%C4%85g_(matematyka)" title="Ciąg (matematyka) – polaco" lang="pl" hreflang="pl" data-title="Ciąg (matematyka)" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Sequensa" title="Sequensa – Piedmontese" lang="pms" hreflang="pms" data-title="Sequensa" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Sequ%C3%AAncia" title="Sequência – portugués" lang="pt" hreflang="pt" data-title="Sequência" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/%C8%98ir_(matematic%C4%83)" title="Șir (matematică) – romanés" lang="ro" hreflang="ro" data-title="Șir (matematică)" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D0%B5%D0%B4%D0%BE%D0%B2%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Последовательность – ruso" lang="ru" hreflang="ru" data-title="Последовательность" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Succissioni_(matimatica)" title="Succissioni (matimatica) – siciliano" lang="scn" hreflang="scn" data-title="Succissioni (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Niz" title="Niz – serbocroata" lang="sh" hreflang="sh" data-title="Niz" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Sequence" title="Sequence – Simple English" lang="en-simple" hreflang="en-simple" data-title="Sequence" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Postupnos%C5%A5_(matematika)" title="Postupnosť (matematika) – eslovaco" lang="sk" hreflang="sk" data-title="Postupnosť (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zaporedje" title="Zaporedje – esloveno" lang="sl" hreflang="sl" data-title="Zaporedje" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Vargu" title="Vargu – albanés" lang="sq" hreflang="sq" data-title="Vargu" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B8%D0%B7" title="Низ – serbio" lang="sr" hreflang="sr" data-title="Низ" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/F%C3%B6ljd" title="Följd – sueco" lang="sv" hreflang="sv" data-title="Följd" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Mfuatano" title="Mfuatano – suahili" lang="sw" hreflang="sw" data-title="Mfuatano" data-language-autonym="Kiswahili" data-language-local-name="suahili" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/%C4%86%C5%AFng_(matymatyka)" title="Ćůng (matymatyka) – Silesian" lang="szl" hreflang="szl" data-title="Ćůng (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%9F%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88" title="தொடர்வரிசை – támil" lang="ta" hreflang="ta" data-title="தொடர்வரிசை" data-language-autonym="தமிழ்" data-language-local-name="támil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A5%E0%B8%B3%E0%B8%94%E0%B8%B1%E0%B8%9A" title="ลำดับ – tailandés" lang="th" hreflang="th" data-title="ลำดับ" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dizi" title="Dizi – turco" lang="tr" hreflang="tr" data-title="Dizi" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%81%D0%BB%D1%96%D0%B4%D0%BE%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Послідовність – ucraíno" lang="uk" hreflang="uk" data-title="Послідовність" data-language-autonym="Українська" data-language-local-name="ucraíno" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%88%D8%A7%D9%84%DB%8C%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="متوالیہ (ریاضی) – urdú" lang="ur" hreflang="ur" data-title="متوالیہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/D%C3%A3y_(to%C3%A1n_h%E1%BB%8Dc)" title="Dãy (toán học) – vietnamita" lang="vi" hreflang="vi" data-title="Dãy (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BA%8F%E5%88%97" title="序列 – chinés wu" lang="wuu" hreflang="wuu" data-title="序列" data-language-autonym="吴语" data-language-local-name="chinés wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-xal mw-list-item"><a href="https://xal.wikipedia.org/wiki/%D0%94%D0%B0%D1%80%D0%B0%D0%BB%D1%82" title="Даралт – calmuco" lang="xal" hreflang="xal" data-title="Даралт" data-language-autonym="Хальмг" data-language-local-name="calmuco" class="interlanguage-link-target"><span>Хальмг</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BA%8F%E5%88%97" title="序列 – chinés" lang="zh" hreflang="zh" data-title="序列" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BA%8F%E5%88%97" title="序列 – cantonés" lang="yue" hreflang="yue" data-title="序列" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q133250#sitelinks-wikipedia" title="Editar as ligazóns interlingüísticas" class="wbc-editpage">Editar as ligazóns</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espazos de nomes"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)" title="Ver o contido da páxina [c]" accesskey="c"><span>Artigo</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Conversa:Sucesi%C3%B3n_(matem%C3%A1ticas)" rel="discussion" title="Conversa acerca do contido desta páxina [t]" accesskey="t"><span>Conversa</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Cambiar a variante de lingua" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">galego</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Vistas"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)"><span>Ler</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit" title="Editar o código fonte desta páxina [e]" accesskey="e"><span>Editar a fonte</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=history" title="Versións anteriores desta páxina [h]" accesskey="h"><span>Ver o historial</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Ferramentas das páxinas"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Ferramentas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Ferramentas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Ferramentas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">agochar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Máis opcións" > <div class="vector-menu-heading"> Accións </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Sucesi%C3%B3n_(matem%C3%A1ticas)"><span>Ler</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit" title="Editar esta páxina [v]" accesskey="v"><span>Editar</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit" title="Editar o código fonte desta páxina [e]" accesskey="e"><span>Editar a fonte</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=history"><span>Ver o historial</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Xeral </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1xinas_que_ligan_con_esta/Sucesi%C3%B3n_(matem%C3%A1ticas)" title="Lista de todas as páxinas do wiki que ligan cara a aquí [j]" accesskey="j"><span>Páxinas que ligan con esta</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:Cambios_relacionados/Sucesi%C3%B3n_(matem%C3%A1ticas)" rel="nofollow" title="Cambios recentes nas páxinas ligadas desde esta [k]" accesskey="k"><span>Cambios relacionados</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1xinas_especiais" title="Lista de todas as páxinas especiais [q]" accesskey="q"><span>Páxinas especiais</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&oldid=6863764" title="Ligazón permanente a esta versión desta páxina"><span>Ligazón permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=info" title="Máis información sobre esta páxina"><span>Información da páxina</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Cita&page=Sucesi%C3%B3n_%28matem%C3%A1ticas%29&id=6863764&wpFormIdentifier=titleform" title="Información sobre como citar esta páxina"><span>Citar esta páxina</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fgl.wikipedia.org%2Fwiki%2FSucesi%25C3%25B3n_%28matem%25C3%25A1ticas%29"><span>Xerar URL acurtado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrKodu&url=https%3A%2F%2Fgl.wikipedia.org%2Fwiki%2FSucesi%25C3%25B3n_%28matem%25C3%25A1ticas%29"><span>Descargar o código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=Sucesi%C3%B3n+%28matem%C3%A1ticas%29"><span>Crear un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Sucesi%C3%B3n_%28matem%C3%A1ticas%29&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&printable=yes" title="Versión para imprimir da páxina [p]" accesskey="p"><span>Versión para imprimir</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Noutros proxectos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Sequence" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q133250" title="Ligazón ao elemento conectado no repositorio de datos [g]" accesskey="g"><span>Elemento de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Ferramentas das páxinas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aparencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aparencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">agochar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-04-1000_artigos_icona_título" class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="//gl.wikipedia.org/wiki/Wikipedia:Wikiproxecto_1000_artigos_de_calidade_para_alumnado_de_12_a_16_anos" title="1000 12/16"><img alt="1000 12/16" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/19px-Hezkuntza_Programa_12-16_ikonoa.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/28px-Hezkuntza_Programa_12-16_ikonoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/38px-Hezkuntza_Programa_12-16_ikonoa.png 2x" data-file-width="747" data-file-height="794" /></a></span></div></div> </div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirección desde «<a href="/w/index.php?title=Sucesi%C3%B3n_matem%C3%A1tica&redirect=no" class="mw-redirect" title="Sucesión matemática">Sucesión matemática</a>»)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="gl" dir="ltr"><p class="mw-empty-elt"> </p><p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, unha <b>sucesión ou secuencia</b> é unha colección enumerada de obxectos na que se permiten as repeticións e importa a <a href="/wiki/Teor%C3%ADa_da_orde" title="Teoría da orde">orde</a>. Como un <a href="/wiki/Conxunto" title="Conxunto">conxunto</a>, contén <a href="/wiki/Elemento_(matem%C3%A1ticas)" title="Elemento (matemáticas)">membros</a> (tamén chamados <i>elementos</i> ou <i>termos</i>). O número de elementos (posiblemente <a href="/wiki/Infinito" title="Infinito">infinito</a> ) chámase <i>lonxitude</i> da secuencia. A diferenza dun conxunto, os mesmos elementos poden aparecer varias veces en diferentes posicións nunha secuencia e, a diferenza dun conxunto, a orde importa. Formalmente, unha secuencia pódese definir como unha <a href="/wiki/Funci%C3%B3n" title="Función">función</a> desde <a href="/wiki/N%C3%BAmero_natural" title="Número natural">os números naturais</a> (as posicións dos elementos na secuencia) ata os elementos en cada posición. A noción de secuencia pódese xeneralizar a unha familia indexada, definida como unha función a partir dun conxunto de índices <i>arbitrario</i>. </p><p>Por exemplo, (M, I, R, E) é unha secuencia de letras coa letra "M" primeiro e "E" por último. Esta secuencia difire de (I, R, M, E). Ademais, a secuencia (1, 1, 2, 3, 5, 8), que contén o número 1 en dúas posicións diferentes, é unha secuencia válida. As secuencias poden ser <i>finitas</i>, como nestes exemplos, ou <i>infinitas</i>, como a secuencia de todos os <a href="/wiki/Paridade_(matem%C3%A1ticas)" title="Paridade (matemáticas)">enteiros</a> <a href="/wiki/N%C3%BAmero_natural" title="Número natural">pares positivos</a> (2, 4, 6, ...). </p><p>A posición dun elemento nunha secuencia é o seu <i>índice</i>. O primeiro elemento ten un índice 0 ou 1, dependendo do contexto ou dunha convención específica. Na <a href="/wiki/An%C3%A1lise_matem%C3%A1tica" title="Análise matemática">análise matemática</a>, unha secuencia adoita denotarse mediante letras en forma de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"></span>, onde o subíndice <i>n</i> refírese ao <i>n</i> ésimo elemento da secuencia; por exemplo, o <i>n-</i>ésimo elemento da <a href="/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci">sucesión de Fibonacci</a> <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span></i> denotase xeralmente como <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cdf519c21deec43f984815e57e15d2dd3575d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}"></span></i>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Cauchy_sequence_illustration2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/350px-Cauchy_sequence_illustration2.svg.png" decoding="async" width="350" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/525px-Cauchy_sequence_illustration2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Cauchy_sequence_illustration2.svg/700px-Cauchy_sequence_illustration2.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption> Unha secuencia infinita de <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a> (en azul). Esta secuencia non é crecente, decrecente, converxente nin <a href="/w/index.php?title=Secuencia_de_Cauchy&action=edit&redlink=1" class="new" title="Secuencia de Cauchy (a páxina aínda non existe)">de Cauchy</a>. No entanto, está limitada.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Notación_e_exemplos"><span id="Notaci.C3.B3n_e_exemplos"></span>Notación e exemplos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=1" title="Editar a sección: «Notación e exemplos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=1" title="Editar o código fonte da sección: Notación e exemplos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Existen varias formas de denotar unha secuencia. Unha forma de especificar unha secuencia é enumerar todos os seus elementos. Por exemplo, os catro primeiros números impares forman a secuencia (1, 3, 5, 7). Esta notación úsase tamén para secuencias infinitas. Por exemplo, a secuencia infinita de enteiros positivos impares escríbese como (1, 3, 5, 7, ...), usando <a href="/wiki/Puntos_suspensivos" title="Puntos suspensivos">puntos suspensivos</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Exemplos">Exemplos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=2" title="Editar a sección: «Exemplos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=2" title="Editar o código fonte da sección: Exemplos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os <a href="/wiki/N%C3%BAmero_primo" title="Número primo">números primos</a> son os <a href="/wiki/N%C3%BAmero_natural" title="Número natural">números naturais</a> maiores que 1 que non teñen <a href="/wiki/Divisor" title="Divisor">divisores</a> senón 1 e eles mesmos. Tomando estes na súa orde natural dáse a secuencia (2, 3, 5, 7, 11, 13, 17, ...). </p><p>Os <a href="/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci">números de Fibonacci</a> comprenden a secuencia enteira cuxos elementos son a suma dos dous elementos anteriores. Os dous primeiros elementos son 0 e 1 ou 1 e 1 polo que temos a secuencia (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...).<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span>[</span>1<span>]</span></a></sup> </p><p>Outros exemplos de secuencias inclúen aquelas formadas por <a href="/wiki/N%C3%BAmero_racional" title="Número racional">números racionais</a>, <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a> e <a href="/wiki/N%C3%BAmero_complexo" title="Número complexo">números complexos</a>. A secuencia (.9, .99, .999, .9999, ...), por exemplo, achégase ao número 1. Outro exemplo pode ser <a href="/wiki/N%C3%BAmero_pi" title="Número pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> como o límite da secuencia (3, 3.1, 3.14, 3.141, 3.1415, ...), que é crecente. Unha secuencia relacionada é a secuencia de díxitos decimais de <span class="texhtml mvar" style="font-style:italic;">π</span>, é dicir, (3, 1, 4, 1, 5, 9, ...). Esta secuencia non ten ningún modelo que sexa facilmente discernible mediante a inspección. </p><p>A <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a> comprende unha gran lista de exemplos de secuencias de enteiros. </p> <div class="mw-heading mw-heading3"><h3 id="Indexación"><span id="Indexaci.C3.B3n"></span>Indexación</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=3" title="Editar a sección: «Indexación»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=3" title="Editar o código fonte da sección: Indexación"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Outras notacións poden ser útiles para secuencias cuxo modelo non se pode adiviñar facilmente ou para secuencias que non teñen un modelo como os díxitos de <a href="/wiki/N%C3%BAmero_pi" title="Número pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a>. Unha destas notacións é escribir unha fórmula xeral para calcular o <i>n-</i>ésimo termo en función de <i>n</i>, encerrala entre parénteses e incluír un subíndice que indique o conxunto de valores que <i>n</i> pode tomar. Por exemplo, nesta notación a secuencia de números pares podería escribirse como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2n)_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (2n)_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3eebc7a123a0c426f026b6e67ee27a1e70024e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.868ex; height:2.843ex;" alt="{\textstyle (2n)_{n\in \mathbb {N} }}"></span>. A secuencia de cadrados podería escribirse como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (n^{2})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (n^{2})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82a392698af06bc8c45da771d3d1fe81fc6e438c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.76ex; height:3.009ex;" alt="{\textstyle (n^{2})_{n\in \mathbb {N} }}"></span>. A variable <i>n</i> chámase <a href="/w/index.php?title=Familia_indexada&action=edit&redlink=1" class="new" title="Familia indexada (a páxina aínda non existe)">índice</a> e o conxunto de valores que pode tomar chámase <a href="/w/index.php?title=Conxunto_de_%C3%ADndices&action=edit&redlink=1" class="new" title="Conxunto de índices (a páxina aínda non existe)">conxunto de índices</a>. </p><p>Unha notaciónn máis pode ser <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8554caa13233f6d842e24c82e18fcdecb70744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="{\textstyle (a_{n})_{n\in \mathbb {N} }}"></span>, que denota unha secuencia cuxo <i>n-</i>ésimo elemento vén dado pola variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span>. Por exemplo: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a_{1}&=1{\text{º elemento de }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{º elemento }}\\a_{3}&=3{\text{º elemento }}\\&\;\;\vdots \\a_{n-1}&=(n-1){\text{-ésimo elemento}}\\a_{n}&=n{\text{-ésimo elemento}}\\a_{n+1}&=(n+1){\text{-ésimo elemento}}\\&\;\;\vdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>º elemento de </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>º elemento </mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>º elemento </mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>-ésimo elemento</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-ésimo elemento</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>-ésimo elemento</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo>⋮</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a_{1}&=1{\text{º elemento de }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{º elemento }}\\a_{3}&=3{\text{º elemento }}\\&\;\;\vdots \\a_{n-1}&=(n-1){\text{-ésimo elemento}}\\a_{n}&=n{\text{-ésimo elemento}}\\a_{n+1}&=(n+1){\text{-ésimo elemento}}\\&\;\;\vdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f16682db2e4891baf81d7088468347272d8626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.671ex; width:32.659ex; height:30.509ex;" alt="{\displaystyle {\begin{aligned}a_{1}&=1{\text{º elemento de }}(a_{n})_{n\in \mathbb {N} }\\a_{2}&=2{\text{º elemento }}\\a_{3}&=3{\text{º elemento }}\\&\;\;\vdots \\a_{n-1}&=(n-1){\text{-ésimo elemento}}\\a_{n}&=n{\text{-ésimo elemento}}\\a_{n+1}&=(n+1){\text{-ésimo elemento}}\\&\;\;\vdots \end{aligned}}}"></span></dd></dl> <p>Pódense considerar varias secuencias ao mesmo tempo empregando diferentes variables; p.ex <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (b_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (b_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c22c52c5a297513aff229e5721a5e3c21abde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.527ex; height:2.843ex;" alt="{\textstyle (b_{n})_{n\in \mathbb {N} }}"></span> podería ser unha secuencia diferente á <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba8554caa13233f6d842e24c82e18fcdecb70744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="{\textstyle (a_{n})_{n\in \mathbb {N} }}"></span>. </p><p>A secuencia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {(a_{n})}_{n=-\infty }^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {(a_{n})}_{n=-\infty }^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6919d3cfe60ad52c41a1cdd35dcc97a9cca33d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.676ex; height:3.176ex;" alt="{\textstyle {(a_{n})}_{n=-\infty }^{\infty }}"></span> é unha <b>secuencia bi-infinita</b>, e tamén se pode escribir como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e26827c279c4c02d239f4ca60f062f94e0820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.227ex; height:2.843ex;" alt="{\textstyle (\ldots ,a_{-1},a_{0},a_{1},a_{2},\ldots )}"></span> . </p><p>Nalgúns casos, os elementos da secuencia están relacionados naturalmente cunha secuencia de enteiros cuxa fórmula pode ser facilmente deducida. Nestes casos, o conxunto de índices pode estar implicado por unha lista dos primeiros elementos abstractos. Por exemplo, a secuencia de cadrados de <a href="/wiki/Paridade_(matem%C3%A1ticas)" title="Paridade (matemáticas)">números impares</a> pódese denotar de calquera das seguintes formas. </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,9,25,\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>25</mn> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,9,25,\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd81efaaf605779aacc5d7092172a8f4d90d61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.284ex; height:2.843ex;" alt="{\displaystyle (1,9,25,\ldots )}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="2em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3fb56a26e33b4528469ae545b02fc1ab57765e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.848ex; height:3.176ex;" alt="{\displaystyle (a_{1},a_{3},a_{5},\ldots ),\qquad a_{k}=k^{2}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(a_{2k-1})}_{k=1}^{\infty },\qquad a_{k}=k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>,</mo> <mspace width="2em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(a_{2k-1})}_{k=1}^{\infty },\qquad a_{k}=k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac14b074b535f5580edcd15776155b62adbd96b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.601ex; height:3.343ex;" alt="{\displaystyle {(a_{2k-1})}_{k=1}^{\infty },\qquad a_{k}=k^{2}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {(a_{k})}_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mo>,</mo> <mspace width="2em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(a_{k})}_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/500c6f9fdda9625ff0481341a071e08eb638af1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.653ex; height:3.343ex;" alt="{\displaystyle {(a_{k})}_{k=1}^{\infty },\qquad a_{k}=(2k-1)^{2}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}(2k-1)^{2}{\bigr )}_{k=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}(2k-1)^{2}{\bigr )}_{k=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e335a2285e6dc7e424dbfcec64654442f2c92a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.559ex; height:3.509ex;" alt="{\displaystyle {\bigl (}(2k-1)^{2}{\bigr )}_{k=1}^{\infty }}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Definición_dunha_secuencia_por_recursividade"><span id="Definici.C3.B3n_dunha_secuencia_por_recursividade"></span>Definición dunha secuencia por recursividade</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=4" title="Editar a sección: «Definición dunha secuencia por recursividade»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=4" title="Editar o código fonte da sección: Definición dunha secuencia por recursividade"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As secuencias cuxos elementos están relacionados cos elementos anteriores dun xeito sinxelo adoitan definirse mediante <a href="/w/index.php?title=Definici%C3%B3n_recursiva&action=edit&redlink=1" class="new" title="Definición recursiva (a páxina aínda non existe)">recursividade</a>. Isto contrasta coa definición de secuencias de elementos en función das súas posicións. </p><p>A <a href="/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci">secuencia de Fibonacci</a> é un exemplo clásico sinxelo, definido pola relación de recorrencia </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=a_{n-1}+a_{n-2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=a_{n-1}+a_{n-2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/575cf7950bcf359b0f03a0f73e15412f598fafbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.132ex; height:2.343ex;" alt="{\displaystyle a_{n}=a_{n-1}+a_{n-2},}"></span></dd></dl> <p>con termos iniciais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f3589226b1f07bd27b7c82d8f470a4685fffe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{0}=0}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f6489d2bc20b48a0f4acb8d102124ef02af3531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{1}=1}"></span>. A partir disto, un simple cálculo mostra que os dez primeiros termos desta secuencia son 0, 1, 1, 2, 3, 5, 8, 13, 21 e 34. </p><p>Unha <i>recorrencia linear con coeficientes constantes</i> é unha relación de recorrencia da forma </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}=c_{0}+c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}=c_{0}+c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122cecc8a7fbd2e502a71ca0fad0843e5f0b2886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.788ex; height:2.343ex;" alt="{\displaystyle a_{n}=c_{0}+c_{1}a_{n-1}+\dots +c_{k}a_{n-k},}"></span></dd></dl> <p>onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0},\dots ,c_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0},\dots ,c_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e1e7c27815ec2e03d2be0a20685c752bee3cc68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.335ex; height:2.009ex;" alt="{\displaystyle c_{0},\dots ,c_{k}}"></span> son <a href="/wiki/Constante_(matem%C3%A1ticas)" title="Constante (matemáticas)">constantes</a>. Existe un método xeral para expresar o termo xeral <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> de tal secuencia en función de <span class="texhtml mvar" style="font-style:italic;">n</span>. No caso da secuencia de Fibonacci, temos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{0}=0,c_{1}=c_{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}=0,c_{1}=c_{2}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3cd8d170b2baebe9532833c9d3e1fd534d899f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.484ex; height:2.509ex;" alt="{\displaystyle c_{0}=0,c_{1}=c_{2}=1,}"></span> e a función resultante de <span class="texhtml mvar" style="font-style:italic;">n</span> vén dada pola <a href="/wiki/Sucesi%C3%B3n_de_Fibonacci" title="Sucesión de Fibonacci">fórmula de Binet</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definición_formal_e_propiedades_básicas"><span id="Definici.C3.B3n_formal_e_propiedades_b.C3.A1sicas"></span>Definición formal e propiedades básicas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=5" title="Editar a sección: «Definición formal e propiedades básicas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=5" title="Editar o código fonte da sección: Definición formal e propiedades básicas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=6" title="Editar a sección: «Definición»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=6" title="Editar o código fonte da sección: Definición"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha <b>sucesión</b> é unha <a href="/wiki/Funci%C3%B3n" title="Función">función</a> definida sobre o conxunto dos <a href="/wiki/N%C3%BAmero_natural" title="Número natural">números naturais</a> agás o cero. É frecuente o uso das letras u, v, w... para designalas, no canto de f, g, h... que serven para as funcións. Do mesmo xeito, a variable denótase normalmente n (por natural) no canto de x, habitual para as variables reais. Por convención, escríbese <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7339b0b08d3045d72ddbacf4a3e50d24640c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle u_{n}}"></span> no canto de u(n): </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:\mathbb {N} *\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∗</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:\mathbb {N} *\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e5fe8bd4cebcd0e0064c9825c7bf5940b744a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.4ex; height:2.176ex;" alt="{\displaystyle u:\mathbb {N} *\rightarrow \mathbb {R} }"></span> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow u_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow u_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ae5afa9245fb081b189ae74a542dd69571504be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.557ex; height:2.176ex;" alt="{\displaystyle n\rightarrow u_{n}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Finita_e_infinita">Finita e infinita</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=7" title="Editar a sección: «Finita e infinita»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=7" title="Editar o código fonte da sección: Finita e infinita"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>lonxitude</b> dunha secuencia defínese como o número de termos da secuencia. </p><p>As secuencias finitas inclúen a <b>secuencia baleira</b> ( ) que non ten elementos. </p><p>Unha secuencia que é infinita en ambas direccións, é dicir, que non ten nin un primeiro nin un elemento final, chámase <b>secuencia bi-infinita ou</b> <b>secuencia infinita bidireccional</b> Unha función do conxunto <b>Z</b> de <i>todos</i> <a href="/wiki/N%C3%BAmero_enteiro" title="Número enteiro">os enteiros</a> nun conxunto, como por exemplo a secuencia de todos os enteiros pares ( ..., −4, −2, 0, 2, 4, 6, 8, ... ), é bi-infinito. Esta secuencia podería ser denotada <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {(2n)}_{n=-\infty }^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {(2n)}_{n=-\infty }^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d4414952cd82fe190386861179f21221ee6305" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.785ex; height:3.176ex;" alt="{\textstyle {(2n)}_{n=-\infty }^{\infty }}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Crecente_e_decrecente">Crecente e decrecente</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=8" title="Editar a sección: «Crecente e decrecente»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=8" title="Editar o código fonte da sección: Crecente e decrecente"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dise que unha secuencia é <i>monótonamente crecente</i> se cada termo é maior ou igual que o anterior. Por exemplo, a secuencia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {(a_{n})}_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {(a_{n})}_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f29610f135bde65faf22368b136917a8bdcdf7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.176ex;" alt="{\textstyle {(a_{n})}_{n=1}^{\infty }}"></span> aumenta monótonamente se e só se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{n+1}\geq a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{n+1}\geq a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee65e9b4bae3bf8d85bed38fb770feaaf66ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.096ex; height:2.343ex;" alt="{\textstyle a_{n+1}\geq a_{n}}"></span> para todos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbf {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbf {N} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8bc5331e84c17aa658e89eba112a11e671d84a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.974ex; height:2.176ex;" alt="{\displaystyle n\in \mathbf {N} .}"></span> Se cada termo consecutivo é estritamente maior que (>) o termo anterior, entón a secuencia chámase <b>estritamente crecente monótonamente</b>. Unha secuencia é <b>decrecente monótonamente</b> se cada termo consecutivo é menor ou igual ao anterior, e é <b>decrecente monótonamente</b> se cada un é estritamente menor que o anterior. Se unha secuencia é crecente ou decrecente chámase secuencia <b>monótona</b>. Este é un caso especial da noción máis xeral dunha función monótona. </p><p>Os termos <b>non decrecente</b> e <b>non crecente</b> úsanse a miúdo en lugar de <i>crecente</i> e <i>decrecente</i> para evitar calquera posible confusión con <i>estritamente crecente</i> e <i>estritamente decrecente</i>, respectivamente. </p> <div class="mw-heading mw-heading3"><h3 id="Limitada">Limitada</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=9" title="Editar a sección: «Limitada»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=9" title="Editar o código fonte da sección: Limitada"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se a secuencia de números reais (<i>a<sub>n</sub></i>) é tal que todos os termos son menores que algún número real <i>M</i>, entón dise que a secuencia está <b>limita superiormente</b>. Noutras palabras, isto significa que existe <i>M</i> tal que para todo <i>n</i>, <i>a <sub>n</sub></i> ≤ <i>M</i>. Calquera tal <i>M</i> denomínase <i>límite superior</i>. Do mesmo xeito, se, para algún <i>m</i> real, <i>un <sub>n</sub></i> ≥ <i>m</i> para todo <i>n</i> maior que algún <i>N</i>, daquela a secuencia está <b>limitada inferiormente</b> e calquera tal <i>m</i> denomínase límite <i>inferior</i>. Se unha secuencia está limitada superiormente e inferiormente, entón dise que a secuencia está <b>limitada</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Subsecuencias">Subsecuencias</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=10" title="Editar a sección: «Subsecuencias»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=10" title="Editar o código fonte da sección: Subsecuencias"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha <b>subsecuencia</b> dunha secuencia dada é unha secuencia formada a partir da secuencia dada eliminando algúns dos elementos sen perturbar as posicións relativas dos elementos restantes. Por exemplo, a secuencia de enteiros pares positivos (2, 4, 6, ...) é unha subsecuencia dos enteiros positivos (1, 2, 3, ...). As posicións dalgúns elementos mudan cando se eliminan outros. Non obstante, consérvanse as posicións relativas. </p><p>Formalmente, unha subsecuencia da secuencia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f0c27000fbc03d920b5b678949a0043ad269bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.759ex; height:2.843ex;" alt="{\displaystyle (a_{n})_{n\in \mathbb {N} }}"></span> é calquera secuencia da forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a_{n_{k}})_{k\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (a_{n_{k}})_{k\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e724316410e8cd4ca71ac1b6dc395d1ce938fae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.489ex; height:3.009ex;" alt="{\textstyle (a_{n_{k}})_{k\in \mathbb {N} }}"></span>, onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n_{k})_{k\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n_{k})_{k\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/800b612131446f0c7286979e24de30ce30dd1ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.664ex; height:2.843ex;" alt="{\displaystyle (n_{k})_{k\in \mathbb {N} }}"></span> é unha secuencia estritamente crecente de enteiros positivos. </p> <div class="mw-heading mw-heading3"><h3 id="Outros_tipos_de_secuencias">Outros tipos de secuencias</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=11" title="Editar a sección: «Outros tipos de secuencias»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=11" title="Editar o código fonte da sección: Outros tipos de secuencias"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Unha <b><a href="/w/index.php?title=Secuencia_de_enteiros&action=edit&redlink=1" class="new" title="Secuencia de enteiros (a páxina aínda non existe)">secuencia de enteiros</a></b> é unha secuencia cuxos termos son enteiros.</li> <li>Unha <b><a href="/w/index.php?title=Sucesi%C3%B3n_polin%C3%B3mica&action=edit&redlink=1" class="new" title="Sucesión polinómica (a páxina aínda non existe)">sucesión polinómica</a></b> é unha sucesión cuxos termos son polinomios.</li> <li>Unha secuencia enteira positiva ás veces chámase <b>multiplicativa</b>, se <i>a</i><sub><i>nm</i></sub> = <i>a</i><sub><i>n</i></sub><i>a</i><sub><i>m</i></sub> para todos os pares <i>n</i>, <i>m</i> tal que <i>n</i> e <i>m</i> son <a href="/wiki/N%C3%BAmeros_primos_entre_si" title="Números primos entre si">coprimos</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>2<span>]</span></a></sup> Noutros casos, as secuencias adoitan chamarse <i>multiplicativas</i>, se <i>a</i><sub><i>n</i></sub> = <i>na</i><sub>1</sub> para todo <i>n</i>.</li> <li>Unha <a href="/wiki/C%C3%B3digo_binario" title="Código binario">secuencia binaria</a> é unha secuencia cuxos termos teñen un de entre dous valores discretos, por exemplo, valores en <a href="/wiki/C%C3%B3digo_binario" title="Código binario">base 2</a> (0,1,1,0, ...), unha serie de lanzamentos de moedas (cara/cruz) C,X,C,C,X,... , as respostas a un conxunto de preguntas Verdadeiro ou Falso (V, F, V, V, ...), etc.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Límites_e_converxencia"><span id="L.C3.ADmites_e_converxencia"></span>Límites e converxencia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=12" title="Editar a sección: «Límites e converxencia»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=12" title="Editar o código fonte da sección: Límites e converxencia"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Converging_Sequence_example.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Converging_Sequence_example.svg/320px-Converging_Sequence_example.svg.png" decoding="async" width="320" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Converging_Sequence_example.svg/480px-Converging_Sequence_example.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Converging_Sequence_example.svg/640px-Converging_Sequence_example.svg.png 2x" data-file-width="586" data-file-height="474" /></a><figcaption> A gráfica dunha secuencia converxente (<i>a<sub>n</sub></i>) móstrase en azul. Na gráfica podemos ver que a secuencia vai converxendo ao límite cero a medida que <i>n</i> aumenta.</figcaption></figure> <p>Se unha secuencia converxe, converxe a un valor particular coñecido como <i>límite</i>. Se unha secuencia converxe a algún límite, entón é <b>converxente</b>. Unha secuencia que non converxe é <b>diverxente</b> (poderían ser mesmo dous límites). </p><p>Por exemplo, a secuencia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{n}={\frac {n+1}{2n^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a_{n}={\frac {n+1}{2n^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a10b786e4f8ba61fd6ca8604e2f538155a807c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:9.47ex; height:4.009ex;" alt="{\textstyle a_{n}={\frac {n+1}{2n^{2}}}}"></span> mostrada na dereita converxe ao valor 0. Por outra banda, as secuencias <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b_{n}=n^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle b_{n}=n^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad8bdc176bda9b6d02addccffefe1655050972d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.763ex; height:2.843ex;" alt="{\textstyle b_{n}=n^{3}}"></span> (que comeza 1, 8, 27, ...) e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}=(-1)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}=(-1)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b03240d43b19c214eda0d5f0947749a9d27a479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.322ex; height:2.843ex;" alt="{\displaystyle c_{n}=(-1)^{n}}"></span> (que comeza −1, 1, −1, 1, ...) ambas as dúas son diverxentes. </p><p>Se unha secuencia converxe, entón o valor ao que converxe é único. Este valor chámase <b>límite</b> da secuencia. O límite dunha sucesión converxente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}"></span> normalmente denótase <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{n\to \infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{n\to \infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3886d813df09cb056ba182f1b71907821d7bb49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.57ex; height:2.509ex;" alt="{\textstyle \lim _{n\to \infty }a_{n}}"></span>. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}"></span> é unha secuencia diverxente, daquela a expresión <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{n\to \infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{n\to \infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3886d813df09cb056ba182f1b71907821d7bb49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.57ex; height:2.509ex;" alt="{\textstyle \lim _{n\to \infty }a_{n}}"></span> carece de sentido. </p> <div class="mw-heading mw-heading3"><h3 id="Definición_formal_de_converxencia"><span id="Definici.C3.B3n_formal_de_converxencia"></span>Definición formal de converxencia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=13" title="Editar a sección: «Definición formal de converxencia»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=13" title="Editar o código fonte da sección: Definición formal de converxencia"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha secuencia de números reais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}"></span> <b>converxe a</b> un número real <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> se, para todos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e04ec3670b50384a3ce48aca42e7cc5131a06b12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.344ex; height:2.176ex;" alt="{\displaystyle \varepsilon >0}"></span>, existe un número natural <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> tal que para todos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b67a4f8e2ce89617f08316bfdcc6f33887b5629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.557ex; height:2.343ex;" alt="{\displaystyle n\geq N}"></span> temos <sup id="cite_ref-Gaughan_3-0" class="reference"><a href="#cite_note-Gaughan-3"><span>[</span>3<span>]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{n}-L|<\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{n}-L|<\varepsilon .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb31ea1110cdc1d2ab0b0e36eecca4b2c29b97d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.994ex; height:2.843ex;" alt="{\displaystyle |a_{n}-L|<\varepsilon .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Aplicacións_e_resultados_importantes"><span id="Aplicaci.C3.B3ns_e_resultados_importantes"></span>Aplicacións e resultados importantes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=14" title="Editar a sección: «Aplicacións e resultados importantes»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=14" title="Editar o código fonte da sección: Aplicacións e resultados importantes"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3872ee8054312f21ff267bc6831745e42b9c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.025ex; height:2.843ex;" alt="{\displaystyle (b_{n})}"></span> son secuencias converxentes, entón existen os seguintes límites e pódense calcular do seguinte xeito: <sup id="cite_ref-Gaughan_3-1" class="reference"><a href="#cite_note-Gaughan-3"><span>[</span>3<span>]</span></a></sup><sup id="cite_ref-Dawkins_4-0" class="reference"><a href="#cite_note-Dawkins-4"><span>[</span>4<span>]</span></a></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>±</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>±</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c66098a892f43850c1d9ff70a543d05060a7cdf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.509ex; height:3.843ex;" alt="{\displaystyle \lim _{n\to \infty }(a_{n}\pm b_{n})=\lim _{n\to \infty }a_{n}\pm \lim _{n\to \infty }b_{n}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>c</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>c</mi> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d857e58f09738fb4feda9b88bf6dd1d4c34ff63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.715ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }ca_{n}=c\lim _{n\to \infty }a_{n}}"></span> para todos os números reais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }(a_{n}b_{n})={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}{\bigl (}\lim _{n\to \infty }b_{n}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }(a_{n}b_{n})={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}{\bigl (}\lim _{n\to \infty }b_{n}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdc1b6a8640c2038688aa7238d3e73d3b8c8ac8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.862ex; height:4.009ex;" alt="{\displaystyle \lim _{n\to \infty }(a_{n}b_{n})={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}{\bigl (}\lim _{n\to \infty }b_{n}{\bigr )}}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\bigl (}\lim \limits _{n\to \infty }a_{n}{\bigr )}{\big /}{\bigl (}\lim \limits _{n\to \infty }b_{n}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo fence="true" stretchy="true" symmetric="true" maxsize="1.2em" minsize="1.2em">/</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\bigl (}\lim \limits _{n\to \infty }a_{n}{\bigr )}{\big /}{\bigl (}\lim \limits _{n\to \infty }b_{n}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe92c3ec81d9090eeacfa94e62b754ef3728d603" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.403ex; height:5.176ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}={\bigl (}\lim \limits _{n\to \infty }a_{n}{\bigr )}{\big /}{\bigl (}\lim \limits _{n\to \infty }b_{n}{\bigr )}}"></span>, sempre que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }b_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }b_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc03e5cfbce2316f7baa360bb110477ff1f8d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.137ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }b_{n}\neq 0}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}^{p}={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}^{p}={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04fa3aa2e42f3ba74d60ea861c8d8b86b5faa89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.89ex; height:4.176ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}^{p}={\bigl (}\lim _{n\to \infty }a_{n}{\bigr )}^{p}}"></span> para todos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dffb51e20581d50c3012634fd9f7b059a68c1c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p>0}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19e309b94a4f0d733334d2cdc304ad38162c9d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.709ex; height:2.509ex;" alt="{\displaystyle a_{n}>0}"></span></li></ul> <ul><li>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\leq b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\leq b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf92d1a0143f27886feeb4e0dfba4011da68319d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.763ex; height:2.509ex;" alt="{\displaystyle a_{n}\leq b_{n}}"></span> para todos os <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> maiores que algún <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span>, daquela<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695f6e7ce5d4ecf6941d8681cd39d6ec9e737f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.082ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}\leq \lim _{n\to \infty }b_{n}}"></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>a<span>]</span></a></sup></li> <li>( Teorema de compresión )<br />Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9666c75720a7978bb33ee8071bfb9b2a19cb8c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.035ex; height:2.843ex;" alt="{\displaystyle (c_{n})}"></span> é unha secuencia tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\leq c_{n}\leq b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\leq c_{n}\leq b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df5564fead0637fd24d96ab5c5601c1e97ce19b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.087ex; height:2.509ex;" alt="{\displaystyle a_{n}\leq c_{n}\leq b_{n}}"></span> para todos os <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6592abd10dbd8e25e84efd66c5f4db57d41fe752" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.557ex; height:2.176ex;" alt="{\displaystyle n>N}"></span> <span style="white-space:nowrap">e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aede7ae0713aa7ff3d7ed19ffbfe1d8487bd112c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.763ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}=L}"></span>,</span><br />daquela <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (c_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9666c75720a7978bb33ee8071bfb9b2a19cb8c99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.035ex; height:2.843ex;" alt="{\displaystyle (c_{n})}"></span> é converxente, e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }c_{n}=L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }c_{n}=L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef5298cee34361ddf7a19bce0d9010ccf6b97356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.566ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }c_{n}=L}"></span> .</li> <li>Se unha secuencia é limitada e monótona, daquela é converxente.</li> <li>Unha secuencia é converxente <a href="/wiki/Se_e_s%C3%B3_se" title="Se e só se">se e só se</a> todas as súas subsecuencias son converxentes.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Secuencias_de_Cauchy">Secuencias de Cauchy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=15" title="Editar a sección: «Secuencias de Cauchy»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=15" title="Editar o código fonte da sección: Secuencias de Cauchy"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Cauchy_sequence_illustration.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/350px-Cauchy_sequence_illustration.svg.png" decoding="async" width="350" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/525px-Cauchy_sequence_illustration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Cauchy_sequence_illustration.svg/700px-Cauchy_sequence_illustration.svg.png 2x" data-file-width="305" data-file-height="170" /></a><figcaption>A gráfica dunha secuencia de Cauchy (<i>X<sub>n</sub></i>), mostrada en azul, cos eixos para <i>X<sub>n</sub></i> e <i>n</i>. Na gráfica, a sucesión está a converxer a un límite. A medida que a distancia entre os termos consecutivos da secuencia se fai menor cando <i>n</i> aumenta. Nos <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a> todas as secuencias de Cauchy converxen a algún límite.</figcaption></figure> <p>Unha sucesión de Cauchy é unha secuencia cuxos termos se achegan arbitrariamente cando n se fai moi grande. A noción de secuencia de Cauchy é importante no estudo de secuencias en <a href="/wiki/Espazo_m%C3%A9trico" title="Espazo métrico">espazos métricos</a> e, en particular, na <a href="/wiki/An%C3%A1lise_real" title="Análise real">análise real</a>. Un resultado particularmente importante na análise real é <i>a caracterización de Cauchy da converxencia para secuencias</i> : </p> <dl><dd>Unha secuencia de números reais é converxente (nos reais) se e só se é Cauchy.</dd></dl> <p>En cambio, hai secuencias de Cauchy de <a href="/wiki/N%C3%BAmero_racional" title="Número racional">números racionais</a> que non son converxentes nos racionais, por exemplo, a secuencia definida por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68209597f05acea5c148021527eb7fe21bd77a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle x_{1}=1}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n+1}={\tfrac {1}{2}}{\bigl (}x_{n}+{\tfrac {2}{x_{n}}}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n+1}={\tfrac {1}{2}}{\bigl (}x_{n}+{\tfrac {2}{x_{n}}}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7a166323b4bbb69a14e40c97df20f73cbc21b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.665ex; height:3.676ex;" alt="{\displaystyle x_{n+1}={\tfrac {1}{2}}{\bigl (}x_{n}+{\tfrac {2}{x_{n}}}{\bigr )}}"></span> é Cauchy, pero non ten límite racional. De forma máis xeral, calquera secuencia de números racionais que converxe a un <a href="/wiki/N%C3%BAmero_irracional" title="Número irracional">número irracional</a> é Cauchy, mais non converxente cando se interpreta como unha secuencia do conxunto de números racionais. </p><p>Os espazos métricos que satisfán a caracterización de converxencia de Cauchy para secuencias chámanse <a href="/wiki/Espazo_m%C3%A9trico_completo" class="mw-redirect" title="Espazo métrico completo">espazos métricos completos</a> e son particularmente apropiados para a análise. </p> <div class="mw-heading mw-heading3"><h3 id="Límites_infinitos"><span id="L.C3.ADmites_infinitos"></span>Límites infinitos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=16" title="Editar a sección: «Límites infinitos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=16" title="Editar o código fonte da sección: Límites infinitos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En cálculo, é común definir a notación para secuencias que non converxen no sentido comentado anteriormente, senón que se fan e permanecen arbitrariamente grandes, ou se fan e permanecen arbitrariamente negativas. Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> faise arbitrariamente grande como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>, escribimos </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}=\infty .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=\infty .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721e78fd32f0b06ed72c74e6712c1a3f94b39f48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.177ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=\infty .}"></span></dd></dl> <p>Neste caso dicimos que a secuencia <b>diverxe</b>. Un exemplo desta secuencia é <span style="white-space:nowrap"><i>a</i><sub><i>n</i></sub> = <i>n</i></span>. </p><p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> vólvese arbitrariamente negativo (é dicir, negativo e grande en magnitude) como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d55d9b32f6fa8fab6a84ea444a6b5a24bb45e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\to \infty }"></span>, escribimos </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}=-\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=-\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ac619f5f8b70cf39037f1cc6c6701aabbcd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.338ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=-\infty }"></span></dd></dl> <p>e dicimos que a secuencia <b>diverxe</b> cara <b>ao infinito negativo</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Serie">Serie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=17" title="Editar a sección: «Serie»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=17" title="Editar o código fonte da sección: Serie"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="detail principal"> <dl><dd><span><span><i>Artigos principais</i><i>:</i> <a href="/wiki/Serie_(matem%C3%A1ticas)" title="Serie (matemáticas)">Serie (matemáticas)</a> e <a href="/wiki/Sumatorio" title="Sumatorio">Sumatorio</a>.</span></span></dd></dl></div> <p>Unha <b>serie</b> é, falando informalmente, a suma dos termos dunha secuencia. É dicir, é unha expresión da forma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d23408336e23cb0903bc2984727f12114ca58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }a_{n}}"></span> ou <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}+a_{2}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}+a_{2}+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17873956e57130ebc58b0653801bfa31707292dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.972ex; height:2.343ex;" alt="{\displaystyle a_{1}+a_{2}+\cdots }"></span>, onde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18bc33c7c35d82b00f88d3a9103ed4738cde41f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.258ex; height:2.843ex;" alt="{\displaystyle (a_{n})}"></span> é unha secuencia de números reais ou complexos. As <b>sumas parciais</b> dunha serie son as expresións resultantes de substituír o símbolo do infinito por un número finito, é dicir, a <i>N</i>-ésima suma parcial da serie. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d23408336e23cb0903bc2984727f12114ca58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }a_{n}}"></span> é o número </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b356346bae67fd23c98b2f2fd63cef3ed82e9d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.884ex; height:7.343ex;" alt="{\displaystyle S_{N}=\sum _{n=1}^{N}a_{n}=a_{1}+a_{2}+\cdots +a_{N}.}"></span></dd></dl> <p>As propias sumas parciais forman unha secuencia <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (S_{N})_{N\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (S_{N})_{N\in \mathbb {N} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cc5d76bffa97a678f4d7942fff2063e75f9901f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.9ex; height:2.843ex;" alt="{\displaystyle (S_{N})_{N\in \mathbb {N} }}"></span>, que se denomina <b>secuencia de sumas parciais</b> da serie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d23408336e23cb0903bc2984727f12114ca58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }a_{n}}"></span>. Se a secuencia de sumas parciais converxe, dicimos que a serie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d23408336e23cb0903bc2984727f12114ca58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.608ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }a_{n}}"></span> é <b>converxente</b>, e o límite <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lim _{N\to \infty }S_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lim _{N\to \infty }S_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66024e65c4208f994051d06621cff947a6cef7ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.711ex; height:2.509ex;" alt="{\textstyle \lim _{N\to \infty }S_{N}}"></span> chámase <b>valor</b> da serie. A mesma notación úsase para indicar unha serie e o seu valor, é dicir, escribimos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }S_{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }S_{N}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d6d2d023b1039716ef4174a185506d52e356c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.417ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }a_{n}=\lim _{N\to \infty }S_{N}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Notas">Notas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=18" title="Editar a sección: «Notas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=18" title="Editar o código fonte da sección: Notas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink"><a href="#cite_ref-:0_1-0">↑</a></span> <span class="reference-text"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20200812220432/https://mathsisfun.com/algebra/sequences-series.html">"Sequences"</a>. <i>www.mathsisfun.com</i>. Arquivado dende <a rel="nofollow" class="external text" href="https://www.mathsisfun.com/algebra/sequences-series.html">o orixinal</a> o 2020-08-12<span class="reference-accessdate">. Consultado o <span class="nowrap">2020-08-17</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3ASucesi%C3%B3n+%28matem%C3%A1ticas%29&rft.atitle=Sequences&rft.genre=unknown&rft.jtitle=www.mathsisfun.com&rft_id=https%3A%2F%2Fwww.mathsisfun.com%2Falgebra%2Fsequences-series.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><cite class="citation book">Lando, Sergei K. (2003-10-21). "7.4 Multiplicative sequences". <i>Lectures on generating functions</i>. AMS. <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:Fontes_bibliogr%C3%A1ficas/978-0-8218-3481-7" title="Especial:Fontes bibliográficas/978-0-8218-3481-7">978-0-8218-3481-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3ASucesi%C3%B3n+%28matem%C3%A1ticas%29&rft.atitle=7.4+Multiplicative+sequences&rft.aufirst=Sergei+K.&rft.aulast=Lando&rft.btitle=Lectures+on+generating+functions&rft.date=2003-10-21&rft.genre=bookitem&rft.isbn=978-0-8218-3481-7&rft.pub=AMS&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Gaughan-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Gaughan_3-0">3,0</a></sup> <sup><a href="#cite_ref-Gaughan_3-1">3,1</a></sup></span> <span class="reference-text"><cite class="citation book">Gaughan, Edward (2009). "1.1 Sequences and Convergence". <i>Introduction to Analysis</i>. AMS (2009). <a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:Fontes_bibliogr%C3%A1ficas/978-0-8218-4787-9" title="Especial:Fontes bibliográficas/978-0-8218-4787-9">978-0-8218-4787-9</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3ASucesi%C3%B3n+%28matem%C3%A1ticas%29&rft.atitle=1.1+Sequences+and+Convergence&rft.aufirst=Edward&rft.aulast=Gaughan&rft.btitle=Introduction+to+Analysis&rft.date=2009&rft.genre=bookitem&rft.isbn=978-0-8218-4787-9&rft.pub=AMS+%282009%29&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Dawkins-4"><span class="mw-cite-backlink"><a href="#cite_ref-Dawkins_4-0">↑</a></span> <span class="reference-text"><cite class="citation web">Dawikins, Paul. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121130095834/http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx">"Series and Sequences"</a>. <i>Paul's Online Math Notes/Calc II (notes)</i>. Arquivado dende <a rel="nofollow" class="external text" href="http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx">o orixinal</a> o 30 November 2012<span class="reference-accessdate">. Consultado o <span class="nowrap">18 December</span> 2012</span>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3ASucesi%C3%B3n+%28matem%C3%A1ticas%29&rft.atitle=Series+and+Sequences&rft.aufirst=Paul&rft.aulast=Dawikins&rft.genre=unknown&rft.jtitle=Paul%27s+Online+Math+Notes%2FCalc+II+%28notes%29&rft_id=http%3A%2F%2Ftutorial.math.lamar.edu%2FClasses%2FCalcII%2FSequences.aspx&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="reflist" style="list-style-type: lower-alpha;"> <ol class="references"> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Se as desigualdades son substituídas por desigualdades estritas, pode non manterse a desigualdade no límite: hai secuencias tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}<b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo><</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}<b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf8e8dce32064580a5c50a9a1569bfb3988bbd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.763ex; height:2.509ex;" alt="{\displaystyle a_{n}<b_{n}}"></span> para todo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, mais <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bcbea4d609a7dddbf58765ca9e1d1db0c31dfbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.082ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }a_{n}=\lim _{n\to \infty }b_{n}}"></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=19" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=19" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table role="presentation" class="mbox-small plainlinks sistersitebox" style="background-color:var(--background-color-neutral-subtle, #f8f9fa);border:1px solid var(--border-color-base, #a2a9b1);color:inherit"> <tbody><tr> <td class="mbox-image"><span class="noviewer" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></td> <td class="mbox-text plainlist"><a href="https://commons.wikimedia.org/wiki/Portada_galega" class="extiw" title="commons:Portada galega">Wikimedia Commons</a> ten máis contidos multimedia na categoría:  <i><b><a href="https://commons.wikimedia.org/wiki/Category:Sequence" class="extiw" title="commons:Category:Sequence">Sucesión</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q133250" title="Modificar a ligazón no Wikidata"><img alt="Modificar a ligazón no Wikidata" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/12px-Arbcom_ru_editing.svg.png" decoding="async" width="12" height="12" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/18px-Arbcom_ru_editing.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Arbcom_ru_editing.svg/24px-Arbcom_ru_editing.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></b></i></td></tr> </tbody></table> <div class="mw-heading mw-heading3"><h3 id="Outros_artigos">Outros artigos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=20" title="Editar a sección: «Outros artigos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=20" title="Editar o código fonte da sección: Outros artigos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></li> <li><a href="/wiki/Relaci%C3%B3n_de_recorrencia" title="Relación de recorrencia">Relación de recorrencia</a></li> <li><a href="/wiki/Progresi%C3%B3n_xeom%C3%A9trica" title="Progresión xeométrica">Progresión xeométrica</a></li> <li><a href="/wiki/Progresi%C3%B3n_aritm%C3%A9tica" title="Progresión aritmética">Progresión aritmética</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Ligazóns_externas"><span id="Ligaz.C3.B3ns_externas"></span>Ligazóns externas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&veaction=edit&section=21" title="Editar a sección: «Ligazóns externas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&action=edit&section=21" title="Editar o código fonte da sección: Ligazóns externas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation book"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Sequence">"Sequence"</a>. <i><a href="/w/index.php?title=Encyclopedia_of_Mathematics&action=edit&redlink=1" class="new" title="Encyclopedia of Mathematics (a páxina aínda non existe)">Encyclopedia of Mathematics</a></i>. <a href="/w/index.php?title=European_Mathematical_Society&action=edit&redlink=1" class="new" title="European Mathematical Society (a páxina aínda non existe)">EMS Press</a>. 2001 [1994].</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fgl.wikipedia.org%3ASucesi%C3%B3n+%28matem%C3%A1ticas%29&rft.atitle=Sequence&rft.btitle=Encyclopedia+of+Mathematics&rft.date=2001&rft.genre=bookitem&rft.pub=EMS+Press&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DSequence&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></li> <li><a rel="nofollow" class="external text" href="http://oeis.org/">The On-Line Encyclopedia of Integer Sequences</a></li> <li><a rel="nofollow" class="external text" href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Journal of Integer Sequences</a> (gratis)</li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; text-align:center;"><a href="/wiki/Axuda:Control_de_autoridades" title="Axuda:Control de autoridades">Control de autoridades</a></th><td class="navbox-list navbox-odd plainlinks" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Wikidata:Main_Page" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/30px-Wikidata-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/40px-Wikidata-logo.svg.png 2x" data-file-width="1050" data-file-height="590" /></a></span>: <span class="uid"><a href="https://www.wikidata.org/wiki/Q133250" class="extiw" title="wikidata:Q133250">Q133250</a></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Art_%26_Architecture_Thesaurus" title="Art & Architecture Thesaurus">AAT</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://vocab.getty.edu/page/aat/300192339">300192339</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_Central_de_Florencia" title="Biblioteca Nacional Central de Florencia">BNCF</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://thes.bncf.firenze.sbn.it/termine.php?id=39374">39374</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_de_Espa%C3%B1a" title="Biblioteca Nacional de España">BNE</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://datos.bne.es/resource/XX533577">XX533577</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_de_Francia" title="Biblioteca Nacional de Francia">BNF</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb121105993">121105993</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Grande_Enciclopedia_Rusa" title="Grande Enciclopedia Rusa">BRE</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://old.bigenc.ru/text/3161778">3161778</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">EBID</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/sequence-mathematics">ID</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Faceted_Application_of_Subject_Terminology" title="Faceted Application of Subject Terminology">FAST</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://id.worldcat.org/fast/1112884">1112884</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4017790-7">4017790-7</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Library_of_Congress_Control_Number" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85120145">sh85120145</a></span></span></span></li> <li><span style="white-space:nowrap;">MAG: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://web.archive.org/web/*/https://academic.microsoft.com/v2/detail/37724570">37724570</a></span></span></span></li> <li><span style="white-space:nowrap;"><a href="/wiki/Biblioteca_Nacional_da_Rep%C3%BAblica_Checa" title="Biblioteca Nacional da República Checa">NKC</a>: <span class="uid"><span class="plainlinks"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph124397">ph124397</a></span></span></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Traído desde «<a dir="ltr" href="https://gl.wikipedia.org/w/index.php?title=Sucesión_(matemáticas)&oldid=6863764">https://gl.wikipedia.org/w/index.php?title=Sucesión_(matemáticas)&oldid=6863764</a>»</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categor%C3%ADas" title="Especial:Categorías">Categoría</a>: <ul><li><a href="/wiki/Categor%C3%ADa:Matem%C3%A1ticas" title="Categoría:Matemáticas">Matemáticas</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorías agochadas: <ul><li><a href="/wiki/Categor%C3%ADa:Wikiproxecto_1000_artigos_de_calidade_para_alumnado_de_12_a_16_anos/Matem%C3%A1ticas" title="Categoría:Wikiproxecto 1000 artigos de calidade para alumnado de 12 a 16 anos/Matemáticas">Wikiproxecto 1000 artigos de calidade para alumnado de 12 a 16 anos/Matemáticas</a></li><li><a href="/wiki/Categor%C3%ADa:Wikipedia:P%C3%A1xinas_con_traduci%C3%B3ns_non_revisadas" title="Categoría:Wikipedia:Páxinas con traducións non revisadas">Wikipedia:Páxinas con traducións non revisadas</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> A última edición desta páxina foi o 15 de outubro de 2024 ás 04:17.</li> <li id="footer-info-copyright">Todo o texto está dispoñible baixo a <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/">licenza Creative Commons recoñecemento compartir igual 4.0</a>; pódense aplicar termos adicionais. Consulte os <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/gl">termos de uso</a> para obter máis información.<br />Wikipedia® é unha marca rexistrada da <a rel="nofollow" class="external text" href="https://www.wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, unha organización sen fins lucrativos.<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Normas de protección de datos</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Acerca_de">Acerca de Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Advertencia_xeral">Advertencias</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Código de conduta</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Desenvolvedores</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/gl.wikipedia.org">Estatísticas</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Declaración de cookies</a></li> <li id="footer-places-mobileview"><a href="//gl.m.wikipedia.org/w/index.php?title=Sucesi%C3%B3n_(matem%C3%A1ticas)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Vista móbil</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5ccf8d5c58-bk9fx","wgBackendResponseTime":168,"wgPageParseReport":{"limitreport":{"cputime":"0.306","walltime":"0.492","ppvisitednodes":{"value":1278,"limit":1000000},"postexpandincludesize":{"value":19298,"limit":2097152},"templateargumentsize":{"value":1313,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":11,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":8849,"limit":5000000},"entityaccesscount":{"value":12,"limit":400},"timingprofile":["100.00% 292.957 1 -total"," 51.27% 150.209 1 Modelo:Control_de_autoridades"," 20.17% 59.097 2 Modelo:Reflist"," 11.92% 34.909 1 Modelo:Cita_web"," 10.59% 31.024 1 Modelo:Commonscat"," 9.96% 29.180 1 Modelo:Irmáns"," 9.28% 27.195 1 Modelo:Caixa_lateral"," 6.71% 19.662 1 Modelo:Springer"," 3.22% 9.440 1 Modelo:1000_artigos_icona_título"," 2.33% 6.815 1 Modelo:Icona_en_título"]},"scribunto":{"limitreport-timeusage":{"value":"0.142","limit":"10.000"},"limitreport-memusage":{"value":6377520,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.canary-6c6bfb5688-rnp6m","timestamp":"20241206093041","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Sucesi\u00f3n (matem\u00e1ticas)","url":"https:\/\/gl.wikipedia.org\/wiki\/Sucesi%C3%B3n_(matem%C3%A1ticas)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q133250","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q133250","author":{"@type":"Organization","name":"Colaboradores dos proxectos da Wikimedia"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2004-05-17T21:57:56Z","dateModified":"2024-10-15T04:17:40Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/7a\/Cauchy_sequence_illustration2.svg","headline":"lista ordenada do mesmo tipo de elementos (finita ou infinita)"}</script> </body> </html>

CINXE.COM

Sucesión (matemáticas) - Wikipedia, a enciclopedia libre