Bernoulli-számok – Wikipédia

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class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-A_Bernoulli-számok_számlálói_és_nevezői" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#A_Bernoulli-számok_számlálói_és_nevezői"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>A Bernoulli-számok számlálói és nevezői</span> </div> </a> <ul id="toc-A_Bernoulli-számok_számlálói_és_nevezői-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aszimptotikus_becslés" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aszimptotikus_becslés"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Aszimptotikus becslés</span> </div> </a> <ul id="toc-Aszimptotikus_becslés-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="Tartalomjegyzék" 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="Tartalomjegyzék kinyitása/becsukása" > <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">Tartalomjegyzék kinyitása/becsukása</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">Bernoulli-számok</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="Ugrás egy más nyelvű szócikkre. Elérhető 36 nyelven" > <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 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Bernoulli_number" title="Bernoulli number – angol" lang="en" hreflang="en" data-title="Bernoulli number" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><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 – arab" lang="ar" hreflang="ar" data-title="عدد برنولي" data-language-autonym="العربية" data-language-local-name="arab" 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 – azerbajdzsáni" lang="az" hreflang="az" data-title="Bernulli ədədləri" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdzsáni" 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 – bolgár" lang="bg" hreflang="bg" data-title="Числа на Бернули" data-language-autonym="Български" data-language-local-name="bolgár" 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 – katalán" lang="ca" hreflang="ca" data-title="Nombres de Bernoulli" data-language-autonym="Català" data-language-local-name="katalán" 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 – cseh" lang="cs" hreflang="cs" data-title="Bernoulliho číslo" data-language-autonym="Čeština" data-language-local-name="cseh" 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 – német" lang="de" hreflang="de" data-title="Bernoulli-Zahl" data-language-autonym="Deutsch" data-language-local-name="német" 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 – görög" lang="el" hreflang="el" data-title="Αριθμός Μπερνούλι" data-language-autonym="Ελληνικά" data-language-local-name="görög" class="interlanguage-link-target"><span>Ελληνικά</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 – spanyol" lang="es" hreflang="es" data-title="Número de Bernoulli" data-language-autonym="Español" data-language-local-name="spanyol" 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 – baszk" lang="eu" hreflang="eu" data-title="Bernoulliren zenbaki" data-language-autonym="Euskara" data-language-local-name="baszk" 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 – perzsa" lang="fa" hreflang="fa" data-title="عدد برنولی" data-language-autonym="فارسی" data-language-local-name="perzsa" 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 – finn" lang="fi" hreflang="fi" data-title="Bernoullin luku" data-language-autonym="Suomi" data-language-local-name="finn" 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 – francia" lang="fr" hreflang="fr" data-title="Nombre de Bernoulli" data-language-autonym="Français" data-language-local-name="francia" 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 – héber" lang="he" hreflang="he" data-title="מספרי ברנולי" data-language-autonym="עברית" data-language-local-name="héber" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numeri_di_Bernoulli" title="Numeri di Bernoulli – olasz" lang="it" hreflang="it" data-title="Numeri di Bernoulli" data-language-autonym="Italiano" data-language-local-name="olasz" 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 – japán" lang="ja" hreflang="ja" data-title="ベルヌーイ数" data-language-autonym="日本語" data-language-local-name="japán" 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 – kazah" lang="kk" hreflang="kk" data-title="Бернулли сандары" data-language-autonym="Қазақша" data-language-local-name="kazah" 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 – koreai" lang="ko" hreflang="ko" data-title="베르누이 수" data-language-autonym="한국어" data-language-local-name="koreai" 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 – lett" lang="lv" hreflang="lv" data-title="Bernulli skaitlis" data-language-autonym="Latviešu" data-language-local-name="lett" 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 – holland" lang="nl" hreflang="nl" data-title="Bernoulligetal" data-language-autonym="Nederlands" data-language-local-name="holland" 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 – norvég (bokmål)" lang="nb" hreflang="nb" data-title="Bernoulli-tall" data-language-autonym="Norsk bokmål" data-language-local-name="norvég (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_Bernoulliego" title="Liczby Bernoulliego – lengyel" lang="pl" hreflang="pl" data-title="Liczby Bernoulliego" data-language-autonym="Polski" data-language-local-name="lengyel" 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 – portugál" lang="pt" hreflang="pt" data-title="Números de Bernoulli" data-language-autonym="Português" data-language-local-name="portugál" 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 – orosz" lang="ru" hreflang="ru" data-title="Числа Бернулли" data-language-autonym="Русский" data-language-local-name="orosz" 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 – szlovén" lang="sl" hreflang="sl" data-title="Bernoullijevo število" data-language-autonym="Slovenščina" data-language-local-name="szlovén" 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 – szerb" lang="sr" hreflang="sr" data-title="Бернулијеви бројеви" data-language-autonym="Српски / srpski" data-language-local-name="szerb" 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 – svéd" lang="sv" hreflang="sv" data-title="Bernoullital" data-language-autonym="Svenska" data-language-local-name="svéd" 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 – török" lang="tr" hreflang="tr" data-title="Bernoulli sayısı" data-language-autonym="Türkçe" data-language-local-name="török" 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 – ukrán" lang="uk" hreflang="uk" data-title="Числа Бернуллі" data-language-autonym="Українська" data-language-local-name="ukrán" 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 – üzbég" lang="uz" hreflang="uz" data-title="Bernoulli sonlari" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="üzbég" 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 – kínai" lang="zh" hreflang="zh" data-title="伯努利数" data-language-autonym="中文" data-language-local-name="kínai" 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 – kantoni" lang="yue" hreflang="yue" data-title="伯努利數" data-language-autonym="粵語" data-language-local-name="kantoni" 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="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások 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id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-indicator-fr-review-status" class="mw-indicator"><indicator name="fr-review-status" class="mw-fr-review-status-indicator" id="mw-fr-revision-toggle"><span class="cdx-fr-css-icon-review--status--stable"></span><b>Ellenőrzött</b></indicator></div> </div> <div id="siteSub" class="noprint">A Wikipédiából, a szabad enciklopédiából</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(<a href="/w/index.php?title=Bernoulli-sz%C3%A1m&redirect=no" class="mw-redirect" title="Bernoulli-szám">Bernoulli-szám</a> szócikkből átirányítva)</span><br /> <div id="mw-fr-revision-messages"><div id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
							cdx-button--size-medium cdx-button--icon-only cdx-dialog__header__close-button" aria-label="Close" onclick="document.getElementById("mw-fr-revision-details").style.display = "none";" type="submit"><span class="cdx-icon cdx-icon--medium
							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">Ez a <a href="/wiki/Wikip%C3%A9dia:Jel%C3%B6lt_lapv%C3%A1ltozatok" title="Wikipédia:Jelölt lapváltozatok">közzétett változat</a>, <a class="external text" href="https://hu.wikipedia.org/w/index.php?title=Speci%C3%A1lis:Rendszernapl%C3%B3k&type=review&page=Bernoulli-sz%C3%A1mok">ellenőrizve</a>: <i>2023. április 6.</i><p><table id="mw-fr-revisionratings-box" class="flaggedrevs-color-1" style="margin: auto;" cellpadding="0"><tr><td class="fr-text" style="vertical-align: middle;">Pontosság</td><td class="fr-value40" style="vertical-align: middle;">ellenőrzött</td></tr></table></p></div></div><div tabindex="0"></div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hu" dir="ltr"><table style="" class="metadata plainlinks ambox ambox-content"> <tbody><tr> <td class="ambox-image mbox-image"> <div class="ambox-image-inner"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td> <td class="ambox-text"><b>Ez a szócikk <a href="/wiki/Wikip%C3%A9dia:T%C3%BCntesd_fel_forr%C3%A1saidat!" title="Wikipédia:Tüntesd fel forrásaidat!"> nem tünteti fel a független forrásokat</a>, amelyeket felhasználtak a készítése során.</b> <small>Emiatt nem tudjuk közvetlenül ellenőrizni, hogy a szócikkben szereplő állítások helytállóak-e. Segíts <a href="/wiki/Wikip%C3%A9dia:T%C3%BCntesd_fel_forr%C3%A1saidat!" title="Wikipédia:Tüntesd fel forrásaidat!">megbízható forrásokat</a> találni az állításokhoz! Lásd még: <a href="/wiki/Wikip%C3%A9dia:A_Wikip%C3%A9dia_nem_az_els%C5%91_k%C3%B6zl%C3%A9s_helye" title="Wikipédia:A Wikipédia nem az első közlés helye">A Wikipédia nem az első közlés helye</a>.</small></td> </tr> </tbody></table> <p>A <b>Bernoulli-számok</b> a <a href="/wiki/Sz%C3%A1melm%C3%A9let" title="Számelmélet">számelméletben</a> előforduló sajátos értékek. A <a href="/wiki/Racion%C3%A1lis_sz%C3%A1mok" title="Racionális számok">racionális számokból</a> álló Bernoulli-számsorozatot <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}(B_{n})_{n\geq 0}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}(B_{n})_{n\geq 0}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef46e4011f8bda14a9753b702f32ed1dcb8ae12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.241ex; height:3.176ex;" alt="{\displaystyle {\bigl (}(B_{n})_{n\geq 0}{\bigr )}}"></span> a következő <a href="/wiki/Rekurzi%C3%B3" title="Rekurzió">rekurzió</a> határozza meg: </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,}"> <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> </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/0703245f3c97022c8a5a0047f6bd3ec57f1b4955" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.726ex; height:2.509ex;" alt="{\displaystyle B_{0}=1,}"></span> továbbá</dd> <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}+2B_{1}=0,}"> <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>2</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}+2B_{1}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f54778260248875533bf2336ac570f4b6003fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.547ex; height:2.509ex;" alt="{\displaystyle B_{0}+2B_{1}=0,}"></span></dd> <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}+3B_{1}+3B_{2}=0,}"> <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>3</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>3</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}+3B_{1}+3B_{2}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d38f9605548aa5b7c1f219349f8579f5fc69d599" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.368ex; height:2.509ex;" alt="{\displaystyle B_{0}+3B_{1}+3B_{2}=0,}"></span></dd> <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}+4B_{1}+6B_{2}+4B_{3}=0,}"> <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>4</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>6</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>4</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}+4B_{1}+6B_{2}+4B_{3}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21035b210e224148a37d2ef17e36b9cb81334062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.189ex; height:2.509ex;" alt="{\displaystyle B_{0}+4B_{1}+6B_{2}+4B_{3}=0,}"></span></dd> <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}+5B_{1}+10B_{2}+10B_{3}+5B_{4}=0,}"> <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>5</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>10</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mn>10</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mn>5</mn> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}+5B_{1}+10B_{2}+10B_{3}+5B_{4}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2c2a1fd5f45b1665adb9f2724548b953397c4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.335ex; height:2.509ex;" alt="{\displaystyle B_{0}+5B_{1}+10B_{2}+10B_{3}+5B_{4}=0,}"></span> és így tovább.</dd> <dd>Általánosan a következő rekurzív képlettel értelmezzük a sorozatot: <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}=-{\frac {1}{n+1}}\cdot \sum _{k=0}^{n-1}{{\Biggl (}{\binom {n+1}{k}}\cdot B_{k}{\Biggr )}}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <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>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">(</mo> </mrow> </mrow> <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>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>⋅</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">)</mo> </mrow> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}=-{\frac {1}{n+1}}\cdot \sum _{k=0}^{n-1}{{\Biggl (}{\binom {n+1}{k}}\cdot B_{k}{\Biggr )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3494b8f5a6d6f3068d8039c107440bbd3458016a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.576ex; height:7.509ex;" alt="{\displaystyle B_{n}=-{\frac {1}{n+1}}\cdot \sum _{k=0}^{n-1}{{\Biggl (}{\binom {n+1}{k}}\cdot B_{k}{\Biggr )}}}"></span>.</dd></dl> <p>Így adódik 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_{0}=1,B_{1}=-{\frac {1}{2}},B_{2}={\frac {1}{6}},B_{3}=0,B_{4}=-{\frac {1}{30}},B_{5}=0,B_{6}={\frac {1}{42}},\dots }"> <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> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>30</mn> </mfrac> </mrow> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>42</mn> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{0}=1,B_{1}=-{\frac {1}{2}},B_{2}={\frac {1}{6}},B_{3}=0,B_{4}=-{\frac {1}{30}},B_{5}=0,B_{6}={\frac {1}{42}},\dots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da0a70e3738180bbe8fba8808db79173cab7f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:68.801ex; height:5.176ex;" alt="{\displaystyle B_{0}=1,B_{1}=-{\frac {1}{2}},B_{2}={\frac {1}{6}},B_{3}=0,B_{4}=-{\frac {1}{30}},B_{5}=0,B_{6}={\frac {1}{42}},\dots }"></span> sorozat. </p><p>A <a href="/wiki/Defin%C3%ADci%C3%B3" title="Definíció">definíció</a> alapján kaphatjuk, hogy teljesül a </p> <center><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 {t}{e^{t}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}t^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {t}{e^{t}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}t^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/475a23828b83ccee3c1f8ae020f6a79d7abf2132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.466ex; height:6.843ex;" alt="{\displaystyle {\frac {t}{e^{t}-1}}=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}t^{n}}"></span></center> <p>sorfejtés. Ebből igazolható, hogy <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_{3}=B_{5}=B_{7}=\cdots =B_{2k+1}=\cdots =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{3}=B_{5}=B_{7}=\cdots =B_{2k+1}=\cdots =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0567fe0ee509d941624adbcd6bb8289c32c9c08a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.43ex; height:2.509ex;" alt="{\displaystyle B_{3}=B_{5}=B_{7}=\cdots =B_{2k+1}=\cdots =0}"></span>. </p><p>A páros indexű Bernoulli-számok a <a href="/wiki/Riemann-f%C3%A9le_z%C3%A9ta-f%C3%BCggv%C3%A9ny" title="Riemann-féle zéta-függvény">Riemann-féle zéta-függvény</a> segítségével is definiálhatóak a következőképpen: </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_{2n}=2(-1)^{n+1}{\frac {\zeta (2n)\;(2n)!}{(2\pi )^{2n}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</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>n</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>n</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</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>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2n}=2(-1)^{n+1}{\frac {\zeta (2n)\;(2n)!}{(2\pi )^{2n}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffe301f5b57eac9c3962e52fdf7c1ca60641d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.767ex; height:6.509ex;" alt="{\displaystyle B_{2n}=2(-1)^{n+1}{\frac {\zeta (2n)\;(2n)!}{(2\pi )^{2n}}}.}"></span></dd></dl> <p>Különféle sorfejtésekben is előfordulnak, például: </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=0}^{m-1}k^{n}={1 \over {n+1}}\sum _{k=0}^{n}{n+1 \choose {k}}B_{k}m^{n+1-k};}"> <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>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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>n</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>n</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> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{m-1}k^{n}={1 \over {n+1}}\sum _{k=0}^{n}{n+1 \choose {k}}B_{k}m^{n+1-k};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b60b1df231d74e2852d654c89a0bfcada66f75f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.634ex; height:7.509ex;" alt="{\displaystyle \sum _{k=0}^{m-1}k^{n}={1 \over {n+1}}\sum _{k=0}^{n}{n+1 \choose {k}}B_{k}m^{n+1-k};}"></span></dd></dl> <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 {\rm {tg}}\left(x\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <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> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {tg}}\left(x\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2693a5a10c71af37f2b18771d21c8cd12f3f83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:43.509ex; height:7.009ex;" alt="{\displaystyle {\rm {tg}}\left(x\right)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}};}"></span></dd></dl> <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 {\rm {ctg}}\left(x\right)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\rm {ctg}}\left(x\right)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f14f6742bb10b060f03f80447f43281a4535732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.425ex; height:7.009ex;" alt="{\displaystyle {\rm {ctg}}\left(x\right)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="A_Bernoulli-számok_számlálói_és_nevezői"><span id="A_Bernoulli-sz.C3.A1mok_sz.C3.A1ml.C3.A1l.C3.B3i_.C3.A9s_nevez.C5.91i"></span>A Bernoulli-számok számlálói és nevezői</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli-sz%C3%A1mok&action=edit&section=1" title="Szakasz szerkesztése: A Bernoulli-számok számlálói és nevezői"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>T. Claussen és C. von Staudt egymástól függetlenül a következő tételt fedezte fel: </p> <dl><dd>Ha <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 m\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e0f3243e9d7f06bee548558bf20aaa9b5263d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 1}"></span>, akkor <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 {\biggl (}B_{2m}+\sum {\frac {1}{p}}{\biggr )}\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msub> <mo>+</mo> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\biggl (}B_{2m}+\sum {\frac {1}{p}}{\biggr )}\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcf12109fa33bb7b1e97634fd233085a183fd343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.661ex; height:6.176ex;" alt="{\displaystyle {\biggl (}B_{2m}+\sum {\frac {1}{p}}{\biggr )}\in \mathbb {Z} }"></span>, ahol azon <i>p</i> prímszámokat összegezzük, amelyekre <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)\mid 2m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∣</mo> <mn>2</mn> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p-1)\mid 2m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb3f9067496997f9509d4cadf914541049400bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.122ex; height:2.843ex;" alt="{\displaystyle (p-1)\mid 2m}"></span>.</dd></dl> <p>Mivel 2-1=1 és 3-1=2 osztója 2-nek, innen azonnal adódik <a href="/wiki/Sr%C3%ADniv%C3%A1sza_R%C3%A1m%C3%A1nudzsan" title="Srínivásza Rámánudzsan">Rámánudzsan</a> észrevétele, hogy ekkor <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_{2m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{2m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3268bfb6b19b0035425fe2cd39461e017a7f05c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.261ex; height:2.509ex;" alt="{\displaystyle B_{2m}}"></span> nevezője osztható 6-tal. </p> <div class="mw-heading mw-heading2"><h2 id="Aszimptotikus_becslés"><span id="Aszimptotikus_becsl.C3.A9s"></span>Aszimptotikus becslés</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Bernoulli-sz%C3%A1mok&action=edit&section=2" title="Szakasz szerkesztése: Aszimptotikus becslés"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>n</i> nagy értékeire érvényes a következő <a href="/wiki/Aszimptotikus_sor" title="Aszimptotikus sor">aszimptotikus</a> formula: <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 \left|{B_{2n}}\right|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo>∼</mo> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>π</mi> <mi>n</mi> </msqrt> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>π</mi> <mi>e</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{B_{2n}}\right|\sim 4{\sqrt {\pi n}}\left({\frac {n}{\pi e}}\right)^{2n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8bbde303f3f61f1d87d76b728a45c182244615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; 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