Número de Bernoulli - Wikipedia, la enciclopedia libre

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class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algunos valores</span> </div> </a> <ul id="toc-Algunos_valores-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identidades_relacionadas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Identidades_relacionadas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Identidades relacionadas</span> </div> </a> <ul id="toc-Identidades_relacionadas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propiedades_aritméticas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propiedades_aritméticas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Propiedades aritméticas</span> </div> </a> <button aria-controls="toc-Propiedades_aritméticas-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Alternar subsección Propiedades aritméticas</span> </button> <ul id="toc-Propiedades_aritméticas-sublist" class="vector-toc-list"> <li id="toc-Continuidad_p-ádica" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuidad_p-ádica"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Continuidad p-ádica</span> </div> </a> <ul id="toc-Continuidad_p-ádica-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fuentes_bibliográficas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fuentes_bibliográficas"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Fuentes bibliográficas</span> </div> </a> <ul id="toc-Fuentes_bibliográficas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_externos-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="Contenidos" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Número de Bernoulli</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ir a un artículo en otro idioma. Disponible en 36 idiomas" > <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-36" 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">36 idiomas</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 (árabe)" lang="ar" hreflang="ar" data-title="عدد برنولي" data-language-autonym="العربية" data-language-local-name="árabe" 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 (azerbaiyano)" lang="az" hreflang="az" data-title="Bernulli ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaiyano" class="interlanguage-link-target"><span>Azərbaycanca</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 (búlgaro)" lang="bg" hreflang="bg" data-title="Числа на Бернули" data-language-autonym="Български" data-language-local-name="búlgaro" 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 (catalán)" lang="ca" hreflang="ca" data-title="Nombres de Bernoulli" data-language-autonym="Català" data-language-local-name="catalán" 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 (checo)" lang="cs" hreflang="cs" data-title="Bernoulliho číslo" data-language-autonym="Čeština" data-language-local-name="checo" 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 (alemán)" lang="de" hreflang="de" data-title="Bernoulli-Zahl" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%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 (griego)" lang="el" hreflang="el" data-title="Αριθμός Μπερνούλι" data-language-autonym="Ελληνικά" data-language-local-name="griego" 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 (inglés)" lang="en" hreflang="en" data-title="Bernoulli number" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Bernoulliren_zenbaki" title="Bernoulliren zenbaki (euskera)" lang="eu" hreflang="eu" data-title="Bernoulliren zenbaki" data-language-autonym="Euskara" data-language-local-name="euskera" 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 (persa)" lang="fa" hreflang="fa" data-title="عدد برنولی" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Bernoullin_luku" title="Bernoullin luku (finés)" lang="fi" hreflang="fi" data-title="Bernoullin luku" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_de_Bernoulli" title="Nombre de Bernoulli (francés)" lang="fr" hreflang="fr" data-title="Nombre de Bernoulli" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</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 (hebreo)" lang="he" hreflang="he" data-title="מספרי ברנולי" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Bernoulli-sz%C3%A1mok" title="Bernoulli-számok (húngaro)" lang="hu" hreflang="hu" data-title="Bernoulli-számok" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</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 (italiano)" lang="it" hreflang="it" data-title="Numeri di Bernoulli" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%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 (japonés)" lang="ja" hreflang="ja" data-title="ベルヌーイ数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</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 (kazajo)" lang="kk" hreflang="kk" data-title="Бернулли сандары" data-language-autonym="Қазақша" data-language-local-name="kazajo" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B2%A0%EB%A5%B4%EB%88%84%EC%9D%B4_%EC%88%98" title="베르누이 수 (coreano)" lang="ko" hreflang="ko" data-title="베르누이 수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Bernulli_skaitlis" title="Bernulli skaitlis (letón)" lang="lv" hreflang="lv" data-title="Bernulli skaitlis" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Bernoulligetal" title="Bernoulligetal (neerlandés)" lang="nl" hreflang="nl" data-title="Bernoulligetal" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Bernoulli-tall" title="Bernoulli-tall (noruego bokmal)" lang="nb" hreflang="nb" data-title="Bernoulli-tall" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Bernoulliego" title="Liczby Bernoulliego (polaco)" lang="pl" hreflang="pl" data-title="Liczby Bernoulliego" data-language-autonym="Polski" data-language-local-name="polaco" 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 (portugués)" lang="pt" hreflang="pt" data-title="Números de Bernoulli" data-language-autonym="Português" data-language-local-name="portugués" 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 (ruso)" lang="ru" hreflang="ru" data-title="Числа Бернулли" data-language-autonym="Русский" data-language-local-name="ruso" 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 (esloveno)" lang="sl" hreflang="sl" data-title="Bernoullijevo število" data-language-autonym="Slovenščina" data-language-local-name="esloveno" 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 (serbio)" lang="sr" hreflang="sr" data-title="Бернулијеви бројеви" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bernoullital" title="Bernoullital (sueco)" lang="sv" hreflang="sv" data-title="Bernoullital" data-language-autonym="Svenska" data-language-local-name="sueco" 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 (tailandés)" lang="th" hreflang="th" data-title="จำนวนแบร์นูลลี" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Bernoulli_say%C4%B1s%C4%B1" title="Bernoulli sayısı (turco)" lang="tr" hreflang="tr" data-title="Bernoulli sayısı" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%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 (ucraniano)" lang="uk" hreflang="uk" data-title="Числа Бернуллі" data-language-autonym="Українська" data-language-local-name="ucraniano" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Bernoulli_sonlari" title="Bernoulli sonlari (uzbeko)" lang="uz" hreflang="uz" data-title="Bernoulli sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 (chino)" lang="zh" hreflang="zh" data-title="伯努利数" data-language-autonym="中文" data-language-local-name="chino" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BC%AF%E5%8A%AA%E5%88%A9%E6%95%B8" title="伯努利數 (cantonés)" lang="yue" hreflang="yue" data-title="伯努利數" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q694114#sitelinks-wikipedia" title="Editar enlaces interlingüísticos" 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height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Commons-emblem-question_book_orange.svg/60px-Commons-emblem-question_book_orange.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Commons-emblem-question_book_orange.svg/80px-Commons-emblem-question_book_orange.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td> <td class="ambox-text"><div class="ambox-text-div"><strong>Este artículo o sección necesita <a href="/wiki/Wikipedia:VER" class="mw-redirect" title="Wikipedia:VER">referencias</a> que aparezcan en una <a href="/wiki/Wikipedia:FF" class="mw-redirect" title="Wikipedia:FF">publicación acreditada</a>.</strong> <span class="hide-when-compact"><br /> <span style="font-size: smaller;"><i>Busca fuentes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="http://www.google.com/search?as_eq=wikipedia&q=%22N%C3%BAmero+de+Bernoulli%22&num=50">«Número de Bernoulli»</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&&as_src=-newswire+-wire+-presswire+-PR+-press+-release+-wikipedia&q=%22N%C3%BAmero+de+Bernoulli%22">noticias</a> · <a rel="nofollow" class="external text" href="http://books.google.com/books?as_brr=0&as_pub=-icon&q=%22N%C3%BAmero+de+Bernoulli%22">libros</a> · <a rel="nofollow" class="external text" href="http://scholar.google.com/scholar?q=%22N%C3%BAmero+de+Bernoulli%22">académico</a> · <a rel="nofollow" class="external text" href="http://images.google.com/images?safe=off&as_rights=(cc_publicdomain%7ccc_attribute%7ccc_sharealike%7ccc_noncommercial%7ccc_nonderived)&q=%22N%C3%BAmero+de+Bernoulli%22">imágenes</a></span></span></span></div><div class="hide-when-compact"><small><div>Este aviso fue puesto el 6 de enero de 2014.</div></small></div></td> </tr> </tbody></table> <p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, los <b>números de Bernoulli</b> (denotados por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"></span> y, a veces, por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"></span> con el fin de distinguirlos de los <a href="/wiki/N%C3%BAmero_de_Bell" title="Número de Bell">números de Bell</a>) constituyen una sucesión de números racionales con profundas conexiones en <a href="/wiki/Teor%C3%ADa_de_n%C3%BAmeros" title="Teoría de números">teoría de números</a>. </p><p>Fueron llamados así por <a href="/wiki/Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a>, en honor de <a href="/wiki/Jacob_Bernoulli" title="Jacob Bernoulli">Jacob Bernoulli</a>, primer matemático que los estudió. Los números de Bernoulli también aparecen en la expansión de las funciones <a href="/wiki/Tangente_(trigonometr%C3%ADa)" title="Tangente (trigonometría)">tangente</a> y <a href="/wiki/Tangente_hiperb%C3%B3lica" title="Tangente hiperbólica">tangente hiperbólica</a> mediante <a href="/wiki/Serie_de_Taylor" title="Serie de Taylor">series de Taylor</a>, en la <a href="/wiki/F%C3%B3rmula_de_Euler-Maclaurin" title="Fórmula de Euler-Maclaurin">fórmula de Euler-Maclaurin</a> y en las expresiones de ciertos valores de la <a href="/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann">función zeta de Riemann</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introducción"><span id="Introducci.C3.B3n"></span>Introducción</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=1" title="Editar sección: Introducción"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Históricamente, surgieron de los trabajos de obtener una <a href="/wiki/F%C3%B3rmula" class="mw-disambig" title="Fórmula">fórmula</a> de la suma de potencias de números naturales, en función de la cantidad de sumandos: </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 _{m=1}^{n}m^{p}=1^{p}+2^{p}+\ldots +n^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{n}m^{p}=1^{p}+2^{p}+\ldots +n^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5022878fcaa67d1bc9ce34fa775a8a82a53ad2e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.27ex; height:6.843ex;" alt="{\displaystyle \sum _{m=1}^{n}m^{p}=1^{p}+2^{p}+\ldots +n^{p}}"></span></dd></dl> <p>Las formas cerradas de la expresión son siempre <a href="/wiki/Polinomio" title="Polinomio">polinomios</a> en <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> de orden <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885ec01d3b5670fd5f88847f32da2b3dd62f60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p+1}"></span>. Se obtuvo una de dichas formas mediante <i><a href="/wiki/Polinomios_de_Bernoulli" title="Polinomios de Bernoulli">polinomios de Bernoulli</a></i> y otra mediante el uso de números de Bernoulli: </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 _{m=1}^{n}m^{p}={\frac {1}{p+1}}\sum _{m=0}^{p}{p+1 \choose m}B_{m}\cdot (n+1)^{p+1-m}=p!\sum _{m=0}^{p}{\frac {B_{m}\cdot (n+1)^{p+1-m}}{(p+1-m)!\,\,m!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mi>p</mi> <mo>!</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>m</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{n}m^{p}={\frac {1}{p+1}}\sum _{m=0}^{p}{p+1 \choose m}B_{m}\cdot (n+1)^{p+1-m}=p!\sum _{m=0}^{p}{\frac {B_{m}\cdot (n+1)^{p+1-m}}{(p+1-m)!\,\,m!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61529e3cafc2e16b9123a19737c38a21795508a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:75.528ex; height:7.009ex;" alt="{\displaystyle \sum _{m=1}^{n}m^{p}={\frac {1}{p+1}}\sum _{m=0}^{p}{p+1 \choose m}B_{m}\cdot (n+1)^{p+1-m}=p!\sum _{m=0}^{p}{\frac {B_{m}\cdot (n+1)^{p+1-m}}{(p+1-m)!\,\,m!}}}"></span></dd></dl> <p><br /> </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 _{m=0}^{n}m^{p}={\frac {B_{p+1}(n+1)-B_{p+1}(0)}{p+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}m^{p}={\frac {B_{p+1}(n+1)-B_{p+1}(0)}{p+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2717a4a9c6c6a87b18462ff77a40075d11a884c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.831ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}m^{p}={\frac {B_{p+1}(n+1)-B_{p+1}(0)}{p+1}}}"></span></dd></dl> <ul><li>Y los polinomios de Bernoulli se pueden calcular a partir de la siguiente fórmula:</li></ul> <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_{p}(x)=\sum _{m=0}^{p}(-1)^{m}{p \choose m}B_{m}\cdot x^{p-m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>p</mi> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p}(x)=\sum _{m=0}^{p}(-1)^{m}{p \choose m}B_{m}\cdot x^{p-m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8213ef5c495375d1a6a4423260061ac014281a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.749ex; height:7.009ex;" alt="{\displaystyle B_{p}(x)=\sum _{m=0}^{p}(-1)^{m}{p \choose m}B_{m}\cdot x^{p-m}}"></span></dd></dl> <p>Donde los <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_{m}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ecf71b60e9a0de2604b6d78e64f4504b739514b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.509ex;" alt="{\displaystyle B_{m}\,}"></span> son los números de Bernouilli, con <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\,}"> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f19ad5528351beae58b5f3e6767c7cfd1934df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.466ex; height:2.509ex;" alt="{\displaystyle B_{0}=1\,}"></span>; los demás números se calculan mediante la siguiente fórmula recursiva: </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_{m}=-{\frac {1}{m+1}}\sum _{k=0}^{m-1}{m+1 \choose {k}}B_{k}=-m!\sum _{k=0}^{m-1}{\frac {B_{k}}{(m+1-k)!\,\,k!}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mrow> <mi>m</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>m</mi> <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>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{m}=-{\frac {1}{m+1}}\sum _{k=0}^{m-1}{m+1 \choose {k}}B_{k}=-m!\sum _{k=0}^{m-1}{\frac {B_{k}}{(m+1-k)!\,\,k!}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4203e751a0834433722b2e94a9d1502aacc327f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:59.79ex; height:7.509ex;" alt="{\displaystyle B_{m}=-{\frac {1}{m+1}}\sum _{k=0}^{m-1}{m+1 \choose {k}}B_{k}=-m!\sum _{k=0}^{m-1}{\frac {B_{k}}{(m+1-k)!\,\,k!}}}"></span></dd></dl> <p>Por ejemplo, si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02ec10a0181916cff4bb685f2c934c4a775c60a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.907ex; height:2.509ex;" alt="{\displaystyle p=1\,}"></span>, tenemos que: </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 _{m=1}^{n}m=1+2+\ldots +n={\frac {1}{2}}\sum _{m=0}^{1}{2 \choose m}B_{m}\cdot (n+1)^{2-m}={\frac {1}{2}}\left[{2 \choose 0}B_{0}\,(n+1)^{2}+{2 \choose 1}B_{1}\,(n+1)\right]={\frac {1\cdot 1\cdot (n^{2}+2n+1)+2\cdot (-{\frac {1}{2}})\cdot (n+1)}{2}}={\frac {n^{2}+n}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>m</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>2</mn> <mi>m</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>2</mn> <mn>0</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mn>2</mn> <mn>1</mn> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{m=1}^{n}m=1+2+\ldots +n={\frac {1}{2}}\sum _{m=0}^{1}{2 \choose m}B_{m}\cdot (n+1)^{2-m}={\frac {1}{2}}\left[{2 \choose 0}B_{0}\,(n+1)^{2}+{2 \choose 1}B_{1}\,(n+1)\right]={\frac {1\cdot 1\cdot (n^{2}+2n+1)+2\cdot (-{\frac {1}{2}})\cdot (n+1)}{2}}={\frac {n^{2}+n}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef3ac7778e7c97fdfdce6c87a835830063c3930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:150.574ex; height:7.509ex;" alt="{\displaystyle \sum _{m=1}^{n}m=1+2+\ldots +n={\frac {1}{2}}\sum _{m=0}^{1}{2 \choose m}B_{m}\cdot (n+1)^{2-m}={\frac {1}{2}}\left[{2 \choose 0}B_{0}\,(n+1)^{2}+{2 \choose 1}B_{1}\,(n+1)\right]={\frac {1\cdot 1\cdot (n^{2}+2n+1)+2\cdot (-{\frac {1}{2}})\cdot (n+1)}{2}}={\frac {n^{2}+n}{2}}}"></span></dd></dl> <p>El primer algoritmo para la generación automática de números de Bernoulli fue descrito por primera vez por <a href="/wiki/Ada_Byron" class="mw-redirect" title="Ada Byron">Ada Lovelace</a> en sus notas del año 1843 sobre la <a href="/wiki/M%C3%A1quina_anal%C3%ADtica" title="Máquina analítica">máquina analítica</a> de <a href="/wiki/Charles_Babbage" title="Charles Babbage">Charles Babbage</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Definición_de_los_números_de_Bernoulli"><span id="Definici.C3.B3n_de_los_n.C3.BAmeros_de_Bernoulli"></span>Definición de los números de Bernoulli</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=2" title="Editar sección: Definición de los números de Bernoulli"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se pueden definir de diversas formas equivalentes: </p> <ul><li>Como los términos independientes de los <a href="/wiki/Polinomios_de_Bernoulli" title="Polinomios de Bernoulli">polinomios de Bernoulli</a> <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_{p}\,(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p}\,(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fefe05f7ee5b60c8e4c10ccc4c6112fdf1e08338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.349ex; height:3.009ex;" alt="{\displaystyle B_{p}\,(x)}"></span> correspondientes, es decir, <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_{p}=B_{p}(0)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{p}=B_{p}(0)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3edb14bb5eadada00f8beb65264b81c7697ba2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.104ex; height:3.009ex;" alt="{\displaystyle B_{p}=B_{p}(0)\,}"></span></li></ul> <ul><li>Mediante una <a href="/wiki/Funci%C3%B3n_generatriz" title="Función generatriz">función generatriz</a> G(<i>x</i>), en este caso:</li></ul> <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 G(x)={\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}\;:\quad {\mbox{si}}\quad |x|<2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <mspace width="thickmathspace" /> <mo>:</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>si</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(x)={\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}\;:\quad {\mbox{si}}\quad |x|<2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e26c6e4ee0d8f6091517d851e566597113d5cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:45.373ex; height:6.843ex;" alt="{\displaystyle G(x)={\frac {x}{e^{x}-1}}=\sum _{n=0}^{\infty }B_{n}{\frac {x^{n}}{n!}}\;:\quad {\mbox{si}}\quad |x|<2\pi }"></span></dd></dl> <p>donde cada coeficiente <i>B</i><sub><i>n</i></sub> de la <a href="/wiki/Serie_de_Taylor" title="Serie de Taylor">serie de Taylor</a> es el <i>n</i>-ésimo número de Bernoulli. </p> <div class="mw-heading mw-heading2"><h2 id="Algunos_valores">Algunos valores</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=3" title="Editar sección: Algunos valores"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A continuación se ofrecen los primeros números de Bernoulli (las sucesiones completas de numeradores y denominadores en <a href="/wiki/OEIS" title="OEIS">OEIS</a> son, respectivamente, <a href="//oeis.org/A027641" class="extiw" title="oeis:A027641">A027641</a> y <a href="//oeis.org/A027642" class="extiw" title="oeis:A027642">A027642</a>): </p> <table class="wikitable" style="margin:auto;"> <tbody><tr> <th>n </th> <th style="width:50px;">0 </th> <th style="width:50px;">1 </th> <th style="width:50px;">2 </th> <th style="width:50px;">3 </th> <th style="width:50px;">4 </th> <th style="width:50px;">5 </th> <th style="width:50px;">6 </th> <th style="width:50px;">7 </th> <th style="width:50px;">8 </th> <th style="width:50px;">9 </th> <th style="width:50px;">10 </th> <th style="width:50px;">11 </th> <th style="width:80px;">12 </th> <th style="width:50px;">13 </th> <th style="width:80px;">14 </th> <th style="width:50px;">15 </th> <th style="width:80px;">16 </th> <th style="width:50px;">17 </th> <th style="width:80px;">18 </th> <th style="width:50px;">19 </th> <th style="width:80px;">20 </th></tr> <tr style="text-align:center;"> <th><i>B</i><sub><i>n</i></sub> </th> <td>1 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">-1</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">2</span></span></span> </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">1</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">6</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">−1</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">30</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">1</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">42</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">−1</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">30</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">5</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">66</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">−691</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">2730</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">7</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">6</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">−3617</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">510</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">43867</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">798</span></span></span> </td> <td>0 </td> <td><span lang="en" style="white-space:nowrap; font-family:FreeSerif,'Linux Libertine','Times New Roman',serif; font-size:118%"><span class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center"><span style="display:block; line-height:1em; padding:0 0.1em">−174611</span><span style="display:none">/</span><span style="display:block; line-height:1em; padding:0 0.1em; border-top:1px solid">330</span></span></span> </td></tr></tbody></table> <p>Se puede demostrar que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=0}"> <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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7358c944cec4dc747db7b0a6add6aa5cb97d0ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.243ex; height:2.509ex;" alt="{\displaystyle B_{n}=0}"></span> para todo <i>n</i> impar distinto de 1. La peculiar forma del valor de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{12}=-{\frac {691}{2730}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>691</mn> <mn>2730</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{12}=-{\frac {691}{2730}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43198ab57c0e9e600f1edd27374f6da45ff2d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.033ex; height:5.343ex;" alt="{\displaystyle B_{12}=-{\frac {691}{2730}}}"></span> parece señalar que los valores de los números de Bernoulli no tienen una descripción elemental; de hecho, esencialmente son valores de la <a href="/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann">función zeta de Riemann</a> para enteros negativos y están asociados a propiedades profundas de la teoría de los números y, por ello, no se espera que tengan una formulación trivial. Se sabe que la suma de los números de Bernoulli diverge. </p> <div class="mw-heading mw-heading2"><h2 id="Identidades_relacionadas">Identidades relacionadas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=4" title="Editar sección: Identidades relacionadas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> expresó los números de Bernoulli en términos de la función zeta de Riemann con la expresión siguiente: </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_{2k}=2(-1)^{k+1}{\frac {\zeta (2k)\;(2k)!}{(2\pi )^{2k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2k}=2(-1)^{k+1}{\frac {\zeta (2k)\;(2k)!}{(2\pi )^{2k}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b24646a81614943a547132de09f422f21bd293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.494ex; height:6.509ex;" alt="{\displaystyle B_{2k}=2(-1)^{k+1}{\frac {\zeta (2k)\;(2k)!}{(2\pi )^{2k}}}}"></span></dd></dl> <p>Para los valores negativos de <i>k</i> mayores o iguales a uno en la función zeta de Riemann se tiene: </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 (-k)=-{\frac {B_{k+1}}{k+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>k</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>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (-k)=-{\frac {B_{k+1}}{k+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e352d0c7b2647452e9a1201489a647208232e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.881ex; height:5.676ex;" alt="{\displaystyle \zeta (-k)=-{\frac {B_{k+1}}{k+1}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Propiedades_aritméticas"><span id="Propiedades_aritm.C3.A9ticas"></span>Propiedades aritméticas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=5" title="Editar sección: Propiedades aritméticas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Como ya se ha indicado, los números de Bernoulli pueden expresarse en términos de la <a href="/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann">función zeta de Riemann</a>, lo que implica que en esencia, son valores de la función zeta para los enteros negativos. Así, se puede esperar que tengan propiedades aritméticas de índole no trivial, un hecho que fue descubierto por <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a> en sus trabajos sobre el <a href="/wiki/%C3%9Altimo_teorema_de_Fermat" title="Último teorema de Fermat">Último teorema de Fermat</a>. </p><p>Las propiedades de los números de Bernoulli relacionados con su divisibilidad se relacionan con los <a href="/w/index.php?title=Grupos_de_clases_ideal&action=edit&redlink=1" class="new" title="Grupos de clases ideal (aún no redactado)">grupos de clases ideales</a> de <a href="/wiki/Cuerpo_ciclot%C3%B3mico" title="Cuerpo ciclotómico">campos ciclotómicos</a> gracias al teorema de Kummer y se refuerzan gracias al <a href="/wiki/Teorema_de_Herbrand-Ribet" title="Teorema de Herbrand-Ribet">teorema de Herbrand-Ribet</a>; también se relacionan con los <a href="/wiki/Cuerpo_cuadr%C3%A1tico" title="Cuerpo cuadrático">campos cuadráticos</a> gracias a las proposiciones de Ankey-Artin-Chowla. Tienen también conexión con las teorías-K algebraicas; si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7e944bcb1be88e9a6a940638f2adce0ec4211a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.009ex;" alt="{\displaystyle c_{n}}"></span> es el numerador de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {B_{n} \over {2n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {B_{n} \over {2n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af759109cac958cb9eeb7081864c2b8175d19192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.783ex; height:3.343ex;" alt="{\displaystyle \scriptstyle {B_{n} \over {2n}}}"></span>, entonces el orden de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle K_{4n-2}(\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle K_{4n-2}(\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8917842f54265a50ee873b764d61abe8e8371469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.108ex; height:2.176ex;" alt="{\displaystyle \scriptstyle K_{4n-2}(\mathbb {Z} )}"></span> es <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_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -c_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2be16a2f59ca9cb8322cc64466b5c3bfbcb30b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.855ex; height:2.343ex;" alt="{\displaystyle -c_{2n}}"></span> si <i>n</i> es par y <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 2c_{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2c_{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e53392a653bf06a0e4d3697341d762166dbac89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.21ex; height:2.509ex;" alt="{\displaystyle 2c_{2n}}"></span> si <i>n</i> es impar. </p><p>Además, relacionada con la cuestión de la divisibilidad, existe un <a href="/wiki/Teorema_de_von_Staudt%E2%80%93Clausen" title="Teorema de von Staudt–Clausen">teorema (von Staudt-Clausen)</a> que nos indica que si sumamos 1/<i>p</i> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"></span> para todo número primo <i>p</i> tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (p-1)|n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (p-1)|n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2828f7d0f70b0df46066595cf0922540a34cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.65ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (p-1)|n}"></span>, el resultado es un número entero. Este hecho nos permite caracterizar de forma inmediata a los denominadores de los números de Bernoulli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"></span> distintos de cero como el producto de todos los números primos <i>p</i> tales que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle (p-1)|n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle (p-1)|n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2828f7d0f70b0df46066595cf0922540a34cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.65ex; height:2.176ex;" alt="{\displaystyle \scriptstyle (p-1)|n}"></span>. En consecuencia los denominadores están libres de cuadrados y son divisibles por 6. </p><p>Finalmente, otro resultado (la <a href="/w/index.php?title=Conjetura_de_Agoh%E2%80%93Giuga&action=edit&redlink=1" class="new" title="Conjetura de Agoh–Giuga (aún no redactado)">conjetura de Agoh–Giuga</a>) postula que <i>p</i> es un número primo <a href="/wiki/Si_y_solo_si" class="mw-redirect" title="Si y solo si">si y solo si</a> <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 \scriptstyle pB_{p-1}\equiv -1{\pmod {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>p</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle pB_{p-1}\equiv -1{\pmod {p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4ac16cf2849add186282911dfaa983abbbb1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.063ex; width:15.175ex; height:2.343ex;" alt="{\displaystyle \scriptstyle pB_{p-1}\equiv -1{\pmod {p}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Continuidad_p-ádica"><span id="Continuidad_p-.C3.A1dica"></span>Continuidad p-ádica</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=6" title="Editar sección: Continuidad p-ádica"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Una importante propiedad relacionada con la congruencia de los números de Bernoulli es la denominada propiedad de la «continuidad <i>p</i>-ádica». Esta propiedad dice lo siguiente: si <i>b</i>, <i>m</i> y <i>n</i> son enteros positivos tales que <i>m</i> y <i>n</i> no son divisibles por <i>p</i> - 1 y <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 \scriptstyle m\equiv n{\pmod {\varphi (p^{b})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>m</mi> <mo>≡</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle m\equiv n{\pmod {\varphi (p^{b})}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f927fada445c22fc914f3cb2b37a36bd59a4ef30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.916ex; height:2.509ex;" alt="{\displaystyle \scriptstyle m\equiv n{\pmod {\varphi (p^{b})}}}"></span>, donde φ() es la <a href="/wiki/Funci%C3%B3n_%CF%86_de_Euler" title="Función φ de Euler">función φ de Euler</a>, entonces: </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 (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>m</mi> </mfrac> </mrow> <mo>≡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/982f69f12f67747600e19f7ed63627d883ea064f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.688ex; height:5.176ex;" alt="{\displaystyle (1-p^{m-1}){\frac {B_{m}}{m}}\equiv (1-p^{n-1}){\frac {B_{n}}{n}}{\pmod {p^{b}}}.}"></span></dd></dl> <p>Y, puesto que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}=-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> <mo>−</mo> <mi>n</mi> <mi>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}=-n\zeta (1-n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac7c329844ddd614a42468588d4011b82dd07eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.586ex; height:2.843ex;" alt="{\displaystyle B_{n}=-n\zeta (1-n)}"></span>, también puede escribirse como: </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 (1-p^{-u})\zeta (u)\equiv (1-p^{-v})\zeta (v){\pmod {p^{b}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>u</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>v</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-p^{-u})\zeta (u)\equiv (1-p^{-v})\zeta (v){\pmod {p^{b}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69f2b7cf62f7dad669693c61cb814fe82a980c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.878ex; height:3.176ex;" alt="{\displaystyle (1-p^{-u})\zeta (u)\equiv (1-p^{-v})\zeta (v){\pmod {p^{b}}}}"></span></dd></dl> <p>donde <i>u</i> = 1 - <i>m</i> y <i>v</i> = 1 - <i>n</i>, de forma que '<i>u</i> y <i>v</i> ni son positivos ni son congruentes con <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 \scriptstyle 1{\pmod {p-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 1{\pmod {p-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4a752113fd3a0a8fd2846c599504dbcfb00358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.04ex; height:2.176ex;" alt="{\displaystyle \scriptstyle 1{\pmod {p-1}}}"></span>. En esencia, esto lo que nos indica es que la <a href="/wiki/Funci%C3%B3n_zeta_de_Riemann" title="Función zeta de Riemann">función zeta de Riemann</a>, con <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 \scriptstyle 1-p^{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle 1-p^{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581e8793a86c0fa5955152d326a759ec98a91436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.716ex; height:2.009ex;" alt="{\displaystyle \scriptstyle 1-p^{z}}"></span> extraídos de la fórmula del <a href="/wiki/Producto_de_Euler" title="Producto de Euler">producto de Euler</a>, es continua tanto en los <a href="/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico">números <i>p</i>-ádicos</a> como en los números enteros negativos congruentes módulo <i>p</i> - 1 con un <i>a</i> concreto tal que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle a\not \equiv 1{\pmod {p-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>a</mi> <mo>≢</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle a\not \equiv 1{\pmod {p-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e79d2a8abf9586e26241969ce3b14bd6df71e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.189ex; height:2.176ex;" alt="{\displaystyle \scriptstyle a\not \equiv 1{\pmod {p-1}}}"></span>, lo que permite extender el resultado a una función continua <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 _{p}(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{p}(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0cf5983341430f9e18b45c191a32370b43ce6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.975ex; height:3.009ex;" alt="{\displaystyle \zeta _{p}(z)}"></span> para todos los enteros <i>p</i>-ádicos <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 \mathbb {Z} _{p},\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p},\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cffa2a743011831d73330deb3541d87ff585b3d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.643ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p},\,}"></span> en lo que se denomina <a href="/wiki/Funci%C3%B3n_zeta_p-%C3%A1dica" class="mw-redirect" title="Función zeta p-ádica">función zeta p-ádica</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=7" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Polinomios_de_Bernoulli" title="Polinomios de Bernoulli">Polinomios de Bernoulli</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Fuentes_bibliográficas"><span id="Fuentes_bibliogr.C3.A1ficas"></span>Fuentes bibliográficas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=8" title="Editar sección: Fuentes bibliográficas"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Álgebra superior de Hall and Knight</li></ul> <ul><li>Cálculo integral de Maynard Kong Requena</li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=N%C3%BAmero_de_Bernoulli&action=edit&section=9" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.bernoulli.org">The Bernoulli Number Page</a></li> <li><span id="Reference-Mathworld-Número_de_Bernoulli" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/BernoulliNumber.html">«Número de Bernoulli»</a>. En Weisstein, Eric W, ed. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i> <span style="color:var(--color-subtle, #555 );">(en inglés)</span>. <a href="/wiki/Wolfram_Research" title="Wolfram Research">Wolfram Research</a>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3AN%C3%BAmero+de+Bernoulli&rft.atitle=N%C3%BAmero+de+Bernoulli&rft.au=Weisstein%2C+Eric+W&rft.aulast=Weisstein%2C+Eric+W&rft.genre=article&rft.jtitle=MathWorld&rft.pub=Wolfram+Research&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FBernoulliNumber.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control 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Número de Bernoulli - Wikipedia, la enciclopedia libre