CINXE.COM
Bernoullin luku – Wikipedia
<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-sticky-header-enabled vector-toc-available" lang="fi" dir="ltr"> <head> <meta charset="UTF-8"> <title>Bernoullin luku – Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-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-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )fiwikimwclientpreferences=([^;]+)/);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": "fi normal","wgMonthNames":["","tammikuu","helmikuu","maaliskuu","huhtikuu","toukokuu","kesäkuu","heinäkuu","elokuu","syyskuu","lokakuu","marraskuu","joulukuu"],"wgRequestId":"7655e309-d678-4619-be8b-7fe7555c5334","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Bernoullin_luku","wgTitle":"Bernoullin luku","wgCurRevisionId":22832020,"wgRevisionId":22832020,"wgArticleId":102201,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Lukuteoria"],"wgPageViewLanguage":"fi","wgPageContentLanguage":"fi","wgPageContentModel":"wikitext","wgRelevantPageName":"Bernoullin_luku","wgRelevantArticleId":102201,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgFlaggedRevsParams":{"tags":{"accuracy":{"levels":3}}},"wgStableRevisionId":22832020, "wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"fi","pageLanguageDir":"ltr","pageVariantFallbacks":"fi"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q694114","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.hidePersonalSandboxEdits":"ready","ext.gadget.fiwiki_flaggedrevs_css_rcfix":"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.flaggedRevs.basic":"ready","mediawiki.codex.messagebox.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.flaggedRevs.advanced","ext.gadget.publicarttablesort","ext.gadget.ViikonKilpailu","ext.gadget.WikiLovesMonunmets","ext.gadget.ProtectionIndicator","ext.gadget.frwiki_infobox_v3","ext.gadget.linkeddata","ext.gadget.perustiedotwikidatassa","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","oojs-ui.styles.icons-media","oojs-ui-core.icons"];</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=fi&modules=ext.cite.styles%7Cext.flaggedRevs.basic%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediamessages.styles%7Cmediawiki.codex.messagebox.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=fi&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=fi&modules=ext.gadget.fiwiki_flaggedrevs_css_rcfix%2ChidePersonalSandboxEdits&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=fi&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.18"> <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 name="viewport" content="width=1120"> <meta property="og:title" content="Bernoullin luku – Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//fi.m.wikipedia.org/wiki/Bernoullin_luku"> <link rel="alternate" type="application/x-wiki" title="Muokkaa" href="/w/index.php?title=Bernoullin_luku&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 (fi)"> <link rel="EditURI" type="application/rsd+xml" href="//fi.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://fi.wikipedia.org/wiki/Bernoullin_luku"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fi"> <link rel="alternate" type="application/atom+xml" title="Wikipedia-Atom-syöte" href="/w/index.php?title=Toiminnot:Tuoreet_muutokset&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-Bernoullin_luku rootpage-Bernoullin_luku skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Siirry sisältöön</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="Sivusto"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Päävalikko" > <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="Päävalikko" > <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">Päävalikko</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">Päävalikko</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">piilota</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Valikko </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:Etusivu" title="Siirry etusivulle [z]" accesskey="z"><span>Etusivu</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:Tietoja"><span>Tietoja Wikipediasta</span></a></li><li id="n-allarticles" class="mw-list-item"><a href="/wiki/Wikipedia:Selaa_luokittain"><span>Kaikki sivut</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Toiminnot:Satunnainen_sivu" title="Avaa satunnainen sivu [x]" accesskey="x"><span>Satunnainen artikkeli</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Toiminnot:Toimintosivut"><span>Toimintosivut</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Osallistuminen </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Ohje:Sis%C3%A4llys" title="Ohjeita"><span>Ohje</span></a></li><li id="n-Kahvihuone" class="mw-list-item"><a href="/wiki/Wikipedia:Kahvihuone"><span>Kahvihuone</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Wikipedia:Ajankohtaista" title="Taustatietoa tämänhetkisistä tapahtumista"><span>Ajankohtaista</span></a></li><li id="n-Tuoreet-odottavat-muutokset" class="mw-list-item"><a href="//fi.wikipedia.org/wiki/Toiminnot:Tuoreet_muutokset?damaging=&goodfaith=&hideliu=0&hideanons=0&userExpLevel=&hidemyself=0&hidebyothers=0&hidebots=1&hidehumans=0&hidepatrolled=1&hideunpatrolled=0&hideminor=0&hidemajor=0&hidepageedits=0&hidenewpages=0&hidecategorization=1&hideWikibase=1&hidelog=0&highlight=1&goodfaith__verylikelybad_color=c5&goodfaith__likelybad_color=c4&goodfaith__maybebad_color=c3&damaging__verylikelybad_color=c5&damaging__likelybad_color=c4&damaging__maybebad_color=c3"><span>Tuoreet odottavat muutokset</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Toiminnot:Tuoreet_muutokset" title="Luettelo tuoreista muutoksista [r]" accesskey="r"><span>Tuoreet muutokset</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:Etusivu" 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="" src="/static/images/mobile/copyright/wikipedia-tagline-fi.svg" width="120" height="13" style="width: 7.5em; 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/Toiminnot:Haku" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Hae Wikipediasta [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Haku</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="Hae Wikipediasta" aria-label="Hae Wikipediasta" autocapitalize="sentences" title="Hae Wikipediasta [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Toiminnot:Haku"> </div> <button class="cdx-button cdx-search-input__end-button">Hae</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Henkilökohtaiset työkalut"> <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="Ulkoasu"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Muuta sivun kirjainkokoa, leveyttä ja väriä" > <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="Ulkoasu" > <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">Ulkoasu</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=fi.wikipedia.org&uselang=fi" class=""><span>Lahjoitukset</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=Toiminnot:Luo_tunnus&returnto=Bernoullin+luku" title="On suositeltavaa luoda käyttäjätunnus ja kirjautua sisään. Se ei kuitenkaan ole pakollista." class=""><span>Luo tunnus</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=Toiminnot:Kirjaudu_sis%C3%A4%C3%A4n&returnto=Bernoullin+luku" title="On suositeltavaa kirjautua sisään. Se ei kuitenkaan ole pakollista. [o]" accesskey="o" class=""><span>Kirjaudu sisään</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="Lisää valintoja" > <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="Henkilökohtaiset työkalut" > <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">Henkilökohtaiset työkalut</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Käyttäjävalikko" > <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=fi.wikipedia.org&uselang=fi"><span>Lahjoitukset</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Toiminnot:Luo_tunnus&returnto=Bernoullin+luku" title="On suositeltavaa luoda käyttäjätunnus ja kirjautua sisään. Se ei kuitenkaan ole pakollista."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Luo tunnus</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Toiminnot:Kirjaudu_sis%C3%A4%C3%A4n&returnto=Bernoullin+luku" title="On suositeltavaa kirjautua sisään. Se ei kuitenkaan ole pakollista. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Kirjaudu sisään</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"> Sivut kirjautumattomille muokkaajille <a href="/wiki/Wikipedia:Tervetuloa_Wikipediaan" aria-label="Lue lisää muokkaamisesta"><span>lue lisää</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/Toiminnot:Omat_muokkaukset" title="Luettelo tästä IP-osoitteesta tehdyistä muokkauksista [y]" accesskey="y"><span>Muokkaukset</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Toiminnot:Oma_keskustelu" title="Keskustelu tämän IP-osoitteen muokkauksista [n]" accesskey="n"><span>Keskustelu</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"><!-- CentralNotice --></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="Sivusto"> <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="Sisällysluettelo" 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">Sisällysluettelo</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">piilota</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">Johdanto</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <ul id="toc-Historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ensimmäiset_Bernoullin_luvut" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ensimmäiset_Bernoullin_luvut"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Ensimmäiset Bernoullin luvut</span> </div> </a> <ul id="toc-Ensimmäiset_Bernoullin_luvut-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bernoullin_luvut_ja_Riemannin_zeta-funktio" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bernoullin_luvut_ja_Riemannin_zeta-funktio"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Bernoullin luvut ja Riemannin zeta-funktio</span> </div> </a> <ul id="toc-Bernoullin_luvut_ja_Riemannin_zeta-funktio-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lähteet" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lähteet"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lähteet</span> </div> </a> <ul id="toc-Lähteet-sublist" class="vector-toc-list"> </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="Sisällysluettelo" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sisällysluettelo" > <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="Vaihda sisällysluettelo" > <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">Vaihda sisällysluettelo</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">Bernoullin luku</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="Mene artikkeliin toisella kielellä. Saatavilla 38 kielellä" > <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-38" 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">38 kieltä</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/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%86%D9%88%D9%84%D9%8A" title="عدد برنولي — arabia" lang="ar" hreflang="ar" data-title="عدد برنولي" data-language-autonym="العربية" data-language-local-name="arabia" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Bernulli_%C9%99d%C9%99dl%C9%99ri" title="Bernulli ədədləri — azeri" lang="az" hreflang="az" data-title="Bernulli ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="azeri" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_Bernoulli" title="Bilangan Bernoulli — indonesia" lang="id" hreflang="id" data-title="Bilangan Bernoulli" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesia" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%B8" title="Числа на Бернули — bulgaria" lang="bg" hreflang="bg" data-title="Числа на Бернули" data-language-autonym="Български" data-language-local-name="bulgaria" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombres_de_Bernoulli" title="Nombres de Bernoulli — katalaani" lang="ca" hreflang="ca" data-title="Nombres de Bernoulli" data-language-autonym="Català" data-language-local-name="katalaani" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Bernoulliho_%C4%8D%C3%ADslo" title="Bernoulliho číslo — tšekki" lang="cs" hreflang="cs" data-title="Bernoulliho číslo" data-language-autonym="Čeština" data-language-local-name="tšekki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Bernoulli-Zahl" title="Bernoulli-Zahl — saksa" lang="de" hreflang="de" data-title="Bernoulli-Zahl" data-language-autonym="Deutsch" data-language-local-name="saksa" 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%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%9C%CF%80%CE%B5%CF%81%CE%BD%CE%BF%CF%8D%CE%BB%CE%B9" title="Αριθμός Μπερνούλι — kreikka" lang="el" hreflang="el" data-title="Αριθμός Μπερνούλι" data-language-autonym="Ελληνικά" data-language-local-name="kreikka" 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/Bernoulli_number" title="Bernoulli number — englanti" lang="en" hreflang="en" data-title="Bernoulli number" data-language-autonym="English" data-language-local-name="englanti" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_de_Bernoulli" title="Número de Bernoulli — espanja" lang="es" hreflang="es" data-title="Número de Bernoulli" data-language-autonym="Español" data-language-local-name="espanja" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bernoulliren_zenbaki" title="Bernoulliren zenbaki — baski" lang="eu" hreflang="eu" data-title="Bernoulliren zenbaki" data-language-autonym="Euskara" data-language-local-name="baski" 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%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%86%D9%88%D9%84%DB%8C" title="عدد برنولی — persia" lang="fa" hreflang="fa" data-title="عدد برنولی" data-language-autonym="فارسی" data-language-local-name="persia" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli" title="Nombre de Bernoulli — ranska" lang="fr" hreflang="fr" data-title="Nombre de Bernoulli" data-language-autonym="Français" data-language-local-name="ranska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_de_Bernoulli" title="Número de Bernoulli — galicia" lang="gl" hreflang="gl" data-title="Número de Bernoulli" data-language-autonym="Galego" data-language-local-name="galicia" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4_%EC%88%98" title="베르누이 수 — korea" lang="ko" hreflang="ko" data-title="베르누이 수" data-language-autonym="한국어" data-language-local-name="korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B0%E0%A5%8D%E0%A4%A8%E0%A5%82%E0%A4%B2%E0%A5%80_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="बर्नूली संख्या — hindi" lang="hi" hreflang="hi" data-title="बर्नूली संख्या" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_di_Bernoulli" title="Numeri di Bernoulli — italia" lang="it" hreflang="it" data-title="Numeri di Bernoulli" data-language-autonym="Italiano" data-language-local-name="italia" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8%D7%99_%D7%91%D7%A8%D7%A0%D7%95%D7%9C%D7%99" title="מספרי ברנולי — heprea" lang="he" hreflang="he" data-title="מספרי ברנולי" data-language-autonym="עברית" data-language-local-name="heprea" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D0%B8_%D1%81%D0%B0%D0%BD%D0%B4%D0%B0%D1%80%D1%8B" title="Бернулли сандары — kazakki" lang="kk" hreflang="kk" data-title="Бернулли сандары" data-language-autonym="Қазақша" data-language-local-name="kazakki" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bernulli_skaitlis" title="Bernulli skaitlis — latvia" lang="lv" hreflang="lv" data-title="Bernulli skaitlis" data-language-autonym="Latviešu" data-language-local-name="latvia" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bernoulli-sz%C3%A1mok" title="Bernoulli-számok — unkari" lang="hu" hreflang="hu" data-title="Bernoulli-számok" data-language-autonym="Magyar" data-language-local-name="unkari" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bernoulligetal" title="Bernoulligetal — hollanti" lang="nl" hreflang="nl" data-title="Bernoulligetal" data-language-autonym="Nederlands" data-language-local-name="hollanti" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E6%95%B0" title="ベルヌーイ数 — japani" lang="ja" hreflang="ja" data-title="ベルヌーイ数" data-language-autonym="日本語" data-language-local-name="japani" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bernoulli-tall" title="Bernoulli-tall — norjan bokmål" lang="nb" hreflang="nb" data-title="Bernoulli-tall" data-language-autonym="Norsk bokmål" data-language-local-name="norjan bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bernoulli_sonlari" title="Bernoulli sonlari — uzbekki" lang="uz" hreflang="uz" data-title="Bernoulli sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Bernoulliego" title="Liczby Bernoulliego — puola" lang="pl" hreflang="pl" data-title="Liczby Bernoulliego" data-language-autonym="Polski" data-language-local-name="puola" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmeros_de_Bernoulli" title="Números de Bernoulli — portugali" lang="pt" hreflang="pt" data-title="Números de Bernoulli" data-language-autonym="Português" data-language-local-name="portugali" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D0%B8" title="Числа Бернулли — venäjä" lang="ru" hreflang="ru" data-title="Числа Бернулли" data-language-autonym="Русский" data-language-local-name="venäjä" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Bernoulli_number" title="Bernoulli number — Simple English" lang="en-simple" hreflang="en-simple" data-title="Bernoulli number" 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-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Bernoullijevo_%C5%A1tevilo" title="Bernoullijevo število — sloveeni" lang="sl" hreflang="sl" data-title="Bernoullijevo število" data-language-autonym="Slovenščina" data-language-local-name="sloveeni" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%B8%D1%98%D0%B5%D0%B2%D0%B8_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B8" title="Бернулијеви бројеви — serbia" lang="sr" hreflang="sr" data-title="Бернулијеви бројеви" data-language-autonym="Српски / srpski" data-language-local-name="serbia" 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/Bernoullital" title="Bernoullital — ruotsi" lang="sv" hreflang="sv" data-title="Bernoullital" data-language-autonym="Svenska" data-language-local-name="ruotsi" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%88%E0%B8%B3%E0%B8%99%E0%B8%A7%E0%B8%99%E0%B9%81%E0%B8%9A%E0%B8%A3%E0%B9%8C%E0%B8%99%E0%B8%B9%E0%B8%A5%E0%B8%A5%E0%B8%B5" title="จำนวนแบร์นูลลี — thai" lang="th" hreflang="th" data-title="จำนวนแบร์นูลลี" data-language-autonym="ไทย" data-language-local-name="thai" 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/Bernoulli_say%C4%B1s%C4%B1" title="Bernoulli sayısı — turkki" lang="tr" hreflang="tr" data-title="Bernoulli sayısı" data-language-autonym="Türkçe" data-language-local-name="turkki" 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%A7%D0%B8%D1%81%D0%BB%D0%B0_%D0%91%D0%B5%D1%80%D0%BD%D1%83%D0%BB%D0%BB%D1%96" title="Числа Бернуллі — ukraina" lang="uk" hreflang="uk" data-title="Числа Бернуллі" data-language-autonym="Українська" data-language-local-name="ukraina" 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/%D8%A8%D8%B1%D9%86%D9%88%D9%84%DB%8C_%D8%B9%D8%AF%D8%AF" title="برنولی عدد — urdu" lang="ur" hreflang="ur" data-title="برنولی عدد" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E6%95%B8" title="伯努利數 — kantoninkiina" lang="yue" hreflang="yue" data-title="伯努利數" data-language-autonym="粵語" data-language-local-name="kantoninkiina" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E6%95%B0" title="伯努利数 — kiina" lang="zh" hreflang="zh" data-title="伯努利数" data-language-autonym="中文" data-language-local-name="kiina" class="interlanguage-link-target"><span>中文</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/Q694114#sitelinks-wikipedia" title="Muokkaa kieltenvälisiä linkkejä" class="wbc-editpage">Muokkaa linkkejä</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="Nimiavaruudet"> <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/Bernoullin_luku" title="Näytä sisältösivu [c]" accesskey="c"><span>Artikkeli</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Keskustelu:Bernoullin_luku" rel="discussion" title="Keskustele sisällöstä [t]" accesskey="t"><span>Keskustelu</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="Valitse kieliversio" > <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">suomi</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="Näkymät"> <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/Bernoullin_luku"><span>Lue</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&veaction=edit" title="Muokkaa tätä sivua [v]" accesskey="v"><span>Muokkaa</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&action=edit" title="Muokkaa tämän sivun lähdekoodia [e]" accesskey="e"><span>Muokkaa wikitekstiä</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&action=history" title="Sivun aikaisemmat versiot [h]" accesskey="h"><span>Näytä historia</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sivutyökalut"> <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="Työkalut" > <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">Työkalut</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">Työkalut</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">piilota</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Lisää valintoja" > <div class="vector-menu-heading"> Toiminnot </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/Bernoullin_luku"><span>Lue</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&veaction=edit" title="Muokkaa tätä sivua [v]" accesskey="v"><span>Muokkaa</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&action=edit" title="Muokkaa tämän sivun lähdekoodia [e]" accesskey="e"><span>Muokkaa wikitekstiä</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&action=history"><span>Näytä historia</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Yleinen </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Toiminnot:T%C3%A4nne_viittaavat_sivut/Bernoullin_luku" title="Lista sivuista, jotka viittaavat tänne [j]" accesskey="j"><span>Tänne viittaavat sivut</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Toiminnot:Linkitetyt_muutokset/Bernoullin_luku" rel="nofollow" title="Viimeisimmät muokkaukset sivuissa, joille viitataan tältä sivulta [k]" accesskey="k"><span>Linkitettyjen sivujen muutokset</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&oldid=22832020" title="Ikilinkki tämän sivun tähän versioon"><span>Ikilinkki</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&action=info" title="Enemmän tietoa tästä sivusta"><span>Sivun tiedot</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:Viittaus&page=Bernoullin_luku&id=22832020&wpFormIdentifier=titleform" title="Tietoa tämän sivun lainaamisesta"><span>Viitetiedot</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:UrlShortener&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FBernoullin_luku"><span>Lyhennä URL-osoite</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:QrCode&url=https%3A%2F%2Ffi.wikipedia.org%2Fwiki%2FBernoullin_luku"><span>Lataa QR-koodi</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"> Tulosta/vie </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Toiminnot:DownloadAsPdf&page=Bernoullin_luku&action=show-download-screen"><span>Lataa PDF-tiedostona</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Bernoullin_luku&printable=yes" title="Tulostettava versio [p]" accesskey="p"><span>Tulostettava versio</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"> Muissa hankkeissa </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/Category:Bernoulli_numbers" 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/Q694114" title="Linkki yhdistettyyn keskustietovaraston kohteeseen [g]" accesskey="g"><span>Wikidata-kohde</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="Sivutyökalut"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Ulkoasu"> <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">Ulkoasu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">siirrä sivupalkkiin</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">piilota</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> <div id="siteSub" class="noprint">Wikipediasta</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fi" dir="ltr"><p><b>Bernoullin luvut</b> ovat <a href="/wiki/Rationaaliluku" title="Rationaaliluku">rationaalilukujono</a>, jolla on suuri merkitys <a href="/wiki/Lukuteoria" title="Lukuteoria">lukuteoriassa</a>. Ensimmäiset Bernoullin luvut ovat: </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 B_{0}=1,\;\;B_{1}=\pm 1/2,\;\;B_{2}=1/6,\;\;B_{3}=0,\;\;B_{4}=-1/30,\;\;B_{5}=0,\;\;B_{6}=1/42.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>±<!-- ± --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>6</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>30</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>42.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}=1,\;\;B_{1}=\pm 1/2,\;\;B_{2}=1/6,\;\;B_{3}=0,\;\;B_{4}=-1/30,\;\;B_{5}=0,\;\;B_{6}=1/42.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429e43ba0dd4807c3e2488509f3c4854e0af0845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.387ex; height:2.843ex;" alt="{\displaystyle B_{0}=1,\;\;B_{1}=\pm 1/2,\;\;B_{2}=1/6,\;\;B_{3}=0,\;\;B_{4}=-1/30,\;\;B_{5}=0,\;\;B_{6}=1/42.}" /></span></dd></dl> <p>On kaksi eri tapaa määritellä Bernoullin luvut. Määritelmät eroavat vain luvun <i>B</i><sub>1</sub> kohdalla: <i>B</i><sub>1</sub> on joko -<style data-mw-deduplicate="TemplateStyles:r22831833">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> tai <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831833" /><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. Ensimmäisessä tapauksessa on kysymyksessä <b>ensimmäiset Bernoullin luvut</b> ja jälkimmäisessä <b>toiset Bernoullin luvut</b>. Koska Bernoullin luvut, joiden indeksi on pariton ja suurempi kuin 1 ovat nollia, käsitellään monesti vain lukuja <i>B</i><sub>2n</sub>, joita joissakin teksteissä merkitään harhaanjohtavasti <i>B</i><sub>n</sub>, kun itse asiassa tarkoitetaan lukua <i>B</i><sub>2n</sub>. Tämä voi johtaa sekaannuksiin; mikäli on tarpeellista käyttää vain lukuja <i>B</i><sub>2n</sub>, voi käyttää vaihtoehtoista merkintää <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}^{*}=B_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {B}_{n}^{*}=B_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c563602f30bf675659fa9b3b62493ad8a0d87237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.885ex; height:2.843ex;" alt="{\displaystyle {B}_{n}^{*}=B_{2n}}" /></span>. </p><p>Bernoullin luvut esiintyvät tangenttifunktion ja hyperbolisen tangenttifunktion <a href="/wiki/Taylorin_sarja" title="Taylorin sarja">Taylorin sarjakehitelmässä</a>, <i>n</i>:n ensimmäisen kokonaisluvun potenssisummien kaavoissa, <a href="/w/index.php?title=Eulerin-Maclaurinin_kaava&action=edit&redlink=1" class="new" title="Eulerin-Maclaurinin kaava (sivua ei ole)">Eulerin-Maclaurinin kaavassa</a> ja eräiden <a href="/wiki/Riemannin_zeta-funktio" class="mw-redirect" title="Riemannin zeta-funktio">Riemannin zeta-funktion</a> arvojen lausekkeissa. </p><p>Ne on nimetty 1600-luvulla eläneen kehittäjänsä <a href="/wiki/Jakob_Bernoulli" title="Jakob Bernoulli">Jakob Bernoullin</a> mukaan.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoullin_luku&veaction=edit&section=1" title="Muokkaa osiota Historia" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bernoullin_luku&action=edit&section=1" title="Muokkaa osion lähdekoodia: Historia"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bernoullin lukujen keksiminen juontaa juurensa ongelmaan kokonaislukupotenssien summan laskemisesta, mikä on kiinnostanut matemaatikkoja antiikin ajoista lähtien. </p><p>Aluksi tunnettiin tapoja laskea <i>n</i>:n ensimmäisen kokonaisluvun, <i>n</i>:n ensimmäisen neliön ja <i>n</i>:n ensimmäisen kuutioluvun summa, mutta ei varsinaisia kaavoja, vain puhtaasti sanallisia kuvauksia. Varhaisista matemaatikoista tätä ongelmaa ovat miettineet ainakin <a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a>, <a href="/wiki/Arkhimedes" title="Arkhimedes">Arkhimedes</a>, <a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a>, <a href="/w/index.php?title=Abu_Bakr_al-Karajidi&action=edit&redlink=1" class="new" title="Abu Bakr al-Karajidi (sivua ei ole)">Abu Bakr al-Karajidi</a> ja <a href="/wiki/Abu_al-Hasan_ibn_al-Haitham" title="Abu al-Hasan ibn al-Haitham">Abu Ali al-Hasan ibn al-Hasan ibn al-Haytham</a>. </p><p><a href="/wiki/Thomas_Harriot" title="Thomas Harriot">Thomas Harriot</a> (n. 1560–1621) on ilmeisesti ensimmäinen, joka johti ja kirjoitti kaavat symbolisessa muodossa, mutta hän etsi kaavat vain neljänsien potenssien summaan asti. Vuonna 1631 julkaistussa Academia Algebraessa <a href="/w/index.php?title=Johann_Faulhaber&action=edit&redlink=1" class="new" title="Johann Faulhaber (sivua ei ole)">Johann Faulhaber</a> esitti kaavat potenssien summille polynomimuodossa aina 17. potenssiin asti ja mainitsi etsineensä kaavat 25. potenssiin asti. Nämäkin kaavat ovat teoksessa, tosin salakirjoitusmuodossa. <a href="/wiki/Donald_E._Knuth" class="mw-redirect" title="Donald E. Knuth">Donald E. Knuth</a> arvelee olevansa ensimmäinen, joka ratkaisi Faulhaberin koodin ja sanoo kaavojen olleen oikein 23. potenssiin asti. Kuitenkaan Faulhaberkaan ei keksinyt yleistä sääntöä. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:JakobBernoulliSummaePotestatum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/360px-JakobBernoulliSummaePotestatum.png" decoding="async" width="360" height="458" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/JakobBernoulliSummaePotestatum.png/540px-JakobBernoulliSummaePotestatum.png 1.5x, //upload.wikimedia.org/wikipedia/commons/7/74/JakobBernoulliSummaePotestatum.png 2x" data-file-width="576" data-file-height="732" /></a><figcaption>Jakob Bernoullin <i>Summae Potestatum</i>, 1713</figcaption></figure> <p>Sveitsiläinen matemaatikko Jakob Bernoulli (1654-1705) oli ensimmäinen, joka huomasi yksittäisen lukujonon, joka tarjoaisi tavan muodostaa yleisen kaavan potenssisummille. Ensimmäinen julkaisu Bernoullin luvuista on Bernoullin hänen kuolemansa jälkeen, vuonna 1713 julkaistussa Ars Conjectandissa. </p><p>Kuvan otteessa kirjasta näkyy sivun alemmassa puoliskossa Bernoullin löytämä kaava. Siinä käytetyt merkinnät kuitenkin poikkeavat useissa kohdissa nykyisistä. Niinpä Bernoulli käyttää merkintöjä A, B C ja D modernin merkintätavan mukaisista luvuista B<sub>2</sub>, B<sub>4</sub>, B<sub>6</sub> ja B<sub>8</sub>. Lausekkeessa c·c−1·c−2·c−3 esiintyvät pisteet ovat erottamassa kertolaskun termejä c, c-1, c-2 ja c-3; nykyisin se kirjoitettaisiin c·(c−1)·(c−2)·(c−3). Modernia termejä käyttäen kyseessä on siis <a href="/wiki/Laskeva_kertoma" class="mw-redirect" title="Laskeva kertoma">laskeva kertoma</a>, jolle käytetään myös merkintää <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^{\underline {k-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{\underline {k-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792568a98da3f190360f4495b8ba5531d639ff9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.198ex; height:2.676ex;" alt="{\displaystyle c^{\underline {k-1}}}" /></span>. Myöskään kertomamerkintää ei vielä tuolloin käytetty. <a href="/wiki/Integraali" title="Integraali">Integraalin</a> merkki on peräisin <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Leibniziltä</a>, joka käytti sitä merkitsemään summaa. Oheisella sivulla Bernoullin käyttämänä esimerkiksi merkintä <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 \int n^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫<!-- ∫ --></mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int n^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99a1bbd3922fb8bffb40c295780fb58fe9cbb5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.03ex; height:5.676ex;" alt="{\displaystyle \int n^{4}}" /></span> tarkoittaa summaa <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^{4}+2^{4}+...+n^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{4}+2^{4}+...+n^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4b0fb27139c29307074546ea9750aea821df49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.441ex; height:2.843ex;" alt="{\displaystyle 1^{4}+2^{4}+...+n^{4}}" /></span>. Niinpä nykyaikaisin merkinnöin Bernoullin kaava kirjoitettaisiin: </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 \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>+</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad3bc96c357dfa3baafe1119960dba553a7693e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.71ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}k^{c}={\frac {n^{c+1}}{c+1}}+{\frac {1}{2}}n^{c}+\sum _{k=2}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}.}" /></span></dd></dl> <p>Tätä voidaan vielä yksinkertaistaa käyttämällä edellä määriteltyjä <i>toisia Bernoullin lukuja</i>, jolloin <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_{1}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{1}={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c7481b9291a7bc09e77a73bad1e6a80ea4ff01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.915ex; height:5.176ex;" alt="{\displaystyle B_{1}={\frac {1}{2}}}" /></span>. Lisäksi voidaan vielä todeta, että laskeva kertoma <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^{\underline {k-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{\underline {k-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792568a98da3f190360f4495b8ba5531d639ff9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.198ex; height:2.676ex;" alt="{\displaystyle c^{\underline {k-1}}}" /></span> saa arvolla <i>k</i>=0 arvon <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 {\frac {1}{c+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{c+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/836f4676a0e5bed31d18acf16c8ef0cc33cfc048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:5.846ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{c+1}}}" /></span>.</ref><span class="kirjaviite" title="Kirjaviite">R. Graham; D. E. Knuth; O. Patashnik: <i>Concrete Mathematics</i>.  (2. painos, luku 2.51)  Addison-Wesley, 1989.  <a href="/wiki/Toiminnot:Kirjal%C3%A4hteet/0-201-55802-5" title="Toiminnot:Kirjalähteet/0-201-55802-5">ISBN 0-201-55802-5</a> </span></ref> Näin ollen Bernoullin kaava yksinkertaistuu muotoon: </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 \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mo>_<!-- _ --></mo> </munder> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>−<!-- − --></mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94800378ba11b922cadd6795c732684350c35bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:27.199ex; height:7.009ex;" alt="{\displaystyle \sum _{k=1}^{n}k^{c}=\sum _{k=0}^{c}{\frac {B_{k}}{k!}}c^{\underline {k-1}}n^{c-k+1}}" /></span></dd></dl> <p>Oheisella Bernoullin teoksen sivulla on kuitenkin virhe: summan <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 \int n^{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫<!-- ∫ --></mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int n^{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30445414001b6d8e3976537e0f4f58445321bdbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.03ex; height:5.676ex;" alt="{\displaystyle \int n^{9}}" /></span> (eli <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 \textstyle \sum _{k=1}^{n}k^{9}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{k=1}^{n}k^{9}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6083bca54a40c95a8df6812c6901238f95ead91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.296ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{k=1}^{n}k^{9}}" /></span>) lausekkeessa viimeisen termin tulisi olla <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 -{\tfrac {3}{20}}n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>20</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {3}{20}}n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e38b33d36fcb0e951ebf4d0c922e28d21dc8ae2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.737ex; height:3.676ex;" alt="{\displaystyle -{\tfrac {3}{20}}n^{2}}" /></span>, ei <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 -{\tfrac {1}{12}}n^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mstyle> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{12}}n^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca96e1525b32a6766b1977134543c0f397db6871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.737ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{12}}n^{2}}" /></span>. </p><p>Bernoullin tyytyväisyys nopeaan tapaan määrittää kertoimet n ensimmäisen kokonaisluvun c:nsien potenssien summalle, jokaiselle kokonaisluvulle c näkyy seuraavasta kommentista: </p><p>"Tämän taulukon avulla sain laskettua alle puolessa neljännestunnissa, että 1000 ensimmäisen luvun kymmenien potenssien summa on 91409924241424243424241924242500.” </p><p>Bernoullin luvut voidaan määritellä <a href="/w/index.php?title=Generoiva_funktio&action=edit&redlink=1" class="new" title="Generoiva funktio (sivua ei ole)">generoivan funktion</a> avulla:<sup id="cite_ref-WOLF_2-0" class="reference"><a href="#cite_note-WOLF-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</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 {\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee22ff2a52d4a7fd9336f7c610dd1b353eb86eab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.949ex; height:6.843ex;" alt="{\displaystyle {\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}.}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Ensimmäiset_Bernoullin_luvut"><span id="Ensimm.C3.A4iset_Bernoullin_luvut"></span>Ensimmäiset Bernoullin luvut</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoullin_luku&veaction=edit&section=2" title="Muokkaa osiota Ensimmäiset Bernoullin luvut" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bernoullin_luku&action=edit&section=2" title="Muokkaa osion lähdekoodia: Ensimmäiset Bernoullin luvut"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tässä taulukossa on vain Bernoullin luvut B<sub>2n</sub>, koska parittomilla indekseillä B<sub>n</sub> on 0, lukuun ottamatta lukua B<sub>1</sub>, jonka arvo on sopimuksesta riippuen joko -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831833" /><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> tai <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831833" /><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> </p> <table class="wikitable" width="260"> <caption>Ensimmäiset Bernoullin luvut, joilla on parillinen indeksi </caption> <tbody><tr> <th>n</th> <th>B<sub>n</sub> </th></tr> <tr align="center"> <td>0</td> <td>1 </td></tr> <tr align="center"> <td>2</td> <td><style data-mw-deduplicate="TemplateStyles:r22831834">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td></tr> <tr align="center"> <td>4</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">30</span></span>⁠</span> </td></tr> <tr align="center"> <td>6</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">42</span></span>⁠</span> </td></tr> <tr align="center"> <td>8</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">30</span></span>⁠</span> </td></tr> <tr align="center"> <td>10</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">66</span></span>⁠</span> </td></tr> <tr align="center"> <td>12</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">691</span><span class="sr-only">/</span><span class="den">2730</span></span>⁠</span> </td></tr> <tr align="center"> <td>14</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td></tr> <tr align="center"> <td>16</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">3617</span><span class="sr-only">/</span><span class="den">510</span></span>⁠</span> </td></tr> <tr align="center"> <td>18</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">43867</span><span class="sr-only">/</span><span class="den">798</span></span>⁠</span> </td></tr> <tr align="center"> <td>20</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">174611</span><span class="sr-only">/</span><span class="den">330</span></span>⁠</span> </td></tr> <tr align="center"> <td>22</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">854513</span><span class="sr-only">/</span><span class="den">138</span></span>⁠</span> </td></tr> <tr align="center"> <td>24</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">236364091</span><span class="sr-only">/</span><span class="den">2730</span></span>⁠</span> </td></tr> <tr align="center"> <td>26</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">8553103</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td></tr> <tr align="center"> <td>28</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">23749461029</span><span class="sr-only">/</span><span class="den">870</span></span>⁠</span> </td></tr> <tr align="center"> <td>30</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">8615841276005</span><span class="sr-only">/</span><span class="den">14322</span></span>⁠</span> </td></tr> <tr align="center"> <td>32</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">7709321041217</span><span class="sr-only">/</span><span class="den">510</span></span>⁠</span> </td></tr> <tr align="center"> <td>34</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">2577687858367</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td></tr> <tr align="center"> <td>36</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">26315271553053477373</span><span class="sr-only">/</span><span class="den">1919190</span></span>⁠</span> </td></tr> <tr align="center"> <td>38</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">2929993913841559</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span> </td></tr> <tr align="center"> <td>40</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">261082718496449122051</span><span class="sr-only">/</span><span class="den">13530</span></span>⁠</span> </td></tr> <tr align="center"> <td>42</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">1520097643918070802691</span><span class="sr-only">/</span><span class="den">1806</span></span>⁠</span> </td></tr> <tr align="center"> <td>44</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">27833269579301024235023</span><span class="sr-only">/</span><span class="den">690</span></span>⁠</span> </td></tr> <tr align="center"> <td>46</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">596451111593912163277961</span><span class="sr-only">/</span><span class="den">282</span></span>⁠</span> </td></tr> <tr align="center"> <td>48</td> <td>−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">5609403368997817686249127547</span><span class="sr-only">/</span><span class="den">46410</span></span>⁠</span> </td></tr> <tr align="center"> <td>50</td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r22831834" /><span class="sfrac">⁠<span class="tion"><span class="num">495057205241079648212477525</span><span class="sr-only">/</span><span class="den">66</span></span>⁠</span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Bernoullin_luvut_ja_Riemannin_zeta-funktio">Bernoullin luvut ja Riemannin zeta-funktio</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoullin_luku&veaction=edit&section=3" title="Muokkaa osiota Bernoullin luvut ja Riemannin zeta-funktio" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bernoullin_luku&action=edit&section=3" title="Muokkaa osion lähdekoodia: Bernoullin luvut ja Riemannin zeta-funktio"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Tiedosto:BernoulliNumbersByZetaLowRes.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/BernoulliNumbersByZetaLowRes.png/250px-BernoulliNumbersByZetaLowRes.png" decoding="async" width="250" height="170" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/78/BernoulliNumbersByZetaLowRes.png/375px-BernoulliNumbersByZetaLowRes.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/78/BernoulliNumbersByZetaLowRes.png/500px-BernoulliNumbersByZetaLowRes.png 2x" data-file-width="580" data-file-height="395" /></a><figcaption>Bernoullin luvut Riemannin zeta-funktion avulla.</figcaption></figure> <p>Bernoullin luvuilla on yhteyksiä Riemannin zeta-funktion arvoihin, Bernoullin luvut voidaan ilmoittaa Riemannin zeta-funktion avulla </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 B_{n}={(-1)}^{n+1}n\zeta (1-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> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mi>n</mi> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}={(-1)}^{n+1}n\zeta (1-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2689a1c96e40c56f71f29027e4f2b5365414d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.877ex; height:3.343ex;" alt="{\displaystyle B_{n}={(-1)}^{n+1}n\zeta (1-n)}" /></span></dd></dl> <p>Riemannin zeta-funktion arvot <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 \zeta (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325ce1491d9c215b41c78846ca17c21dba8e3fb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.299ex; height:2.843ex;" alt="{\displaystyle \zeta (n)}" /></span> voidaan ilmoittaa yksinkertaisella lausekkeella Bernoullin lukujen avulla, kun n on negatiivinen kokonaisluku tai parillinen positiivinen kokonaisluku: parillisille positiivisille kokonaisluvuille 2<i>n</i> </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 \zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π<!-- π --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e375feab181b7b24bd0f3a8dde98f0cd6f22aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.644ex; height:6.676ex;" alt="{\displaystyle \zeta (2n)=(-1)^{n+1}{\frac {B_{2n}(2\pi )^{2n}}{2(2n)!}}}" /></span></dd></dl> <p>ja negatiivisille kokonaisluvuille pätee </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 \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/745d83a9305e85e80ee607b2d1778f08488584ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.248ex; height:5.509ex;" alt="{\displaystyle \zeta (-n)=-{\frac {B_{n+1}}{n+1}}}" /></span>, kun <span style="white-space:nowrap"><i>n</i> ≥ 1</span>,</dd></dl> <p>Kiinteästä yhteydestä kertoo myös se, että <a href="/wiki/Riemannin_hypoteesi" title="Riemannin hypoteesi">Riemannin hypoteesi</a> voidaan muotoilla uudelleen Bernoullin lukuja käyttäen. <a href="/w/index.php?title=Marcel_Riesz&action=edit&redlink=1" class="new" title="Marcel Riesz (sivua ei ole)">Marcel Riesz</a> todisti vuonna 1916 seuraavan hypoteesin olevan yhtäpitävä Riemannin hypoteesin kanssa: </p> <p style="margin-left:32px">Jokaiselle luvulle <i>ε</i> > 1/4 on olemassa (<i>ε</i>:sta riippuva) vakio <i>C</i><sub><i>ε</i></sub> > 0 siten, että |<i>R</i>(<i>x</i>)| < <i>C</i><sub>ε</sub> <i>x</i><sup>ε</sup> kun <i>x</i> → ∞.</p> <p>Merkintä <i>R</i>(<i>x</i>) tarkoittaa <a href="/w/index.php?title=Rieszin_funktio&action=edit&redlink=1" class="new" title="Rieszin funktio (sivua ei ole)">Rieszin funktiota</a> </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 R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left(B_{2k}/(2k)\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π<!-- π --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π<!-- π --></mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> <msub> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left(B_{2k}/(2k)\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169dd77d9014d51dfd76d47cc7780488776b887a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.842ex; height:7.343ex;" alt="{\displaystyle R(x)=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\left(B_{2k}/(2k)\right)}}=2\sum _{k=1}^{\infty }{\frac {k^{\overline {k}}x^{k}}{(2\pi )^{2k}\beta _{2k}}}.\ }" /></span></dd></dl> <p>Tässä <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^{\overline {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\overline {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78fe35f0bb6113078433821196cdc98562133364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.577ex; height:3.176ex;" alt="{\displaystyle n^{\overline {k}}}" /></span> tarkoittaa <a href="/wiki/Nouseva_kertoma" class="mw-redirect" title="Nouseva kertoma">nousevaa kertomaa</a>, käyttäen <a href="/wiki/Donald_E._Knuth" class="mw-redirect" title="Donald E. Knuth">Donald E. Knuthin</a> esittämää merkintää. </p> <div class="mw-heading mw-heading2"><h2 id="Lähteet"><span id="L.C3.A4hteet"></span>Lähteet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoullin_luku&veaction=edit&section=4" title="Muokkaa osiota Lähteet" class="mw-editsection-visualeditor"><span>muokkaa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Bernoullin_luku&action=edit&section=4" title="Muokkaa osion lähdekoodia: Lähteet"><span>muokkaa wikitekstiä</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r23002391">.mw-parser-output div.viitteet-malline ol.references{list-style-type:inherit}.mw-parser-output .viitteet-sarakkeet li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output ol.references>li:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output sup.reference:target{background-color:var(--background-color-interactive,#DEF)}.mw-parser-output span[rel="mw:referencedBy"]{counter-reset:mw-ref-linkback 0}.mw-parser-output span[rel="mw:referencedBy"]>a::before{font-style:italic;content:counter(mw-ref-linkback,lower-alpha)}.mw-parser-output .mw-ref>a[data-mw-group=lower-roman]::after{content:"["counter(mw-Ref,lower-roman)"]"}.mw-parser-output .mw-ref>a[data-mw-group=lower-greek]::after{content:"["counter(mw-Ref,lower-greek)"]"}.mw-parser-output .mw-ref>a[data-mw-group=upper-roman]::after{content:"["counter(mw-Ref,upper-roman)"]"}.mw-parser-output .mw-ref>a[data-mw-group=decimal]::after{content:"["counter(mw-Ref,decimal)"]"}.mw-parser-output .mw-ref>a[data-mw-group=lower-alpha]::after{content:"["counter(mw-Ref,lower-alpha)"]"}.mw-parser-output .mw-ref>a[data-mw-group=upper-alpha]::after{content:"["counter(mw-Ref,upper-alpha)"]"}body.action-info .mw-parser-output :target{background-color:var(--background-color-interactive,#DEF)}</style><div id="viitteet-malline" class="viitteet-malline" style="list-style-type:decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><span class="verkkoviite" title="Verkkoviite"><a rel="nofollow" class="external text" href="http://www.britannica.com/EBchecked/topic/62599/Jakob-Bernoulli">Jakob Bernoulli</a> Encyclopedia Britannica. Viitattu 6.10.2012.</span></span> </li> <li id="cite_note-WOLF-2"><span class="mw-cite-backlink"><a href="#cite_ref-WOLF_2-0">↑</a></span> <span class="reference-text"><span class="verkkoviite" title="Verkkoviite">Kellner B: <a rel="nofollow" class="external text" href="http://www.bernoulli.org">The Bernoulli Number Page</a> <i>bernoulli.org</i>. Viitattu 6.10.2012.</span></span> </li> </ol> </div></div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><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="">Noudettu kohteesta ”<a dir="ltr" href="https://fi.wikipedia.org/w/index.php?title=Bernoullin_luku&oldid=22832020">https://fi.wikipedia.org/w/index.php?title=Bernoullin_luku&oldid=22832020</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Toiminnot:Luokat" title="Toiminnot:Luokat">Luokka</a>: <ul><li><a href="/wiki/Luokka:Lukuteoria" title="Luokka:Lukuteoria">Lukuteoria</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"> Sivua on viimeksi muutettu 17. marraskuuta 2024 kello 18.24.</li> <li id="footer-info-copyright">Teksti on saatavilla <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.fi">Creative Commons Attribution/Share-Alike</a> -lisenssillä; lisäehtoja voi sisältyä. Katso <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use/fi">käyttöehdot</a>.<br /> Wikipedia® on <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundationin</a> rekisteröimä tavaramerkki.<br /> <a href="/wiki/Wikipedia:Artikkelien_ongelmat" title="Wikipedia:Artikkelien ongelmat">Ongelma artikkelissa?</a></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Tietosuojakäytäntö</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:Tietoja">Tietoja Wikipediasta</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:Vastuuvapaus">Vastuuvapaus</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Käytössäännöstö</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Kehittäjät</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/fi.wikipedia.org">Tilastot</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Evästekäytäntö</a></li> <li id="footer-places-mobileview"><a href="//fi.m.wikipedia.org/w/index.php?title=Bernoullin_luku&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobiilinäkymä</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"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></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"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Haku</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div 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="Hae Wikipediasta"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Toiminnot:Haku"> </div> <button class="cdx-button cdx-search-input__end-button">Hae</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Sisällysluettelo" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Vaihda sisällysluettelo" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-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">Vaihda sisällysluettelo</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Bernoullin luku</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>38 kieltä</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Lisää aihe</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-64b5bb4b79-mzqgf","wgBackendResponseTime":155,"wgPageParseReport":{"limitreport":{"cputime":"0.097","walltime":"0.204","ppvisitednodes":{"value":956,"limit":1000000},"postexpandincludesize":{"value":8237,"limit":2097152},"templateargumentsize":{"value":446,"limit":2097152},"expansiondepth":{"value":7,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":21138,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 81.952 1 -total"," 33.33% 27.315 1 Malline:Viitteet"," 24.19% 19.827 1 Malline:Kirjaviite"," 16.96% 13.899 25 Malline:Sfrac"," 16.12% 13.208 4 Malline:Murtoluku"," 9.67% 7.925 2 Malline:Verkkoviite"," 3.17% 2.601 1 Malline:Nowrap"]},"scribunto":{"limitreport-timeusage":{"value":"0.004","limit":"10.000"},"limitreport-memusage":{"value":690887,"limit":52428800}},"cachereport":{"origin":"mw-web.eqiad.main-7db5fb7dbd-hwmwg","timestamp":"20250225160408","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Bernoullin luku","url":"https:\/\/fi.wikipedia.org\/wiki\/Bernoullin_luku","sameAs":"http:\/\/www.wikidata.org\/entity\/Q694114","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q694114","author":{"@type":"Organization","name":"Wikimedia-hankkeiden muokkaajat"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-12-10T11:07:50Z","headline":"rationaalilukujono"}</script> </body> </html>