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class="vector-toc-numb">2</span> <span>Summability</span> </div> </a> <button aria-controls="toc-Summability-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>Toggle Summability subsection</span> </button> <ul id="toc-Summability-sublist" class="vector-toc-list"> <li id="toc-Heuristics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Heuristics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Heuristics</span> </div> </a> <ul id="toc-Heuristics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zeta_function_regularization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zeta_function_regularization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Zeta function regularization</span> </div> </a> <ul id="toc-Zeta_function_regularization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cutoff_regularization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cutoff_regularization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Cutoff regularization</span> </div> </a> <ul id="toc-Cutoff_regularization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ramanujan_summation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ramanujan_summation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Ramanujan summation</span> </div> </a> <ul id="toc-Ramanujan_summation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Failure_of_stable_linear_summation_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Failure_of_stable_linear_summation_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Failure of stable linear summation methods</span> </div> </a> <ul id="toc-Failure_of_stable_linear_summation_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Physics</span> </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_media" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>In popular media</span> </div> </a> <ul id="toc-In_popular_media-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">1 + 2 + 3 + 4 + ⋯</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" 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Available in 27 languages" > <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-27" 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">27 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 + 2 + 3 + 4 + · · · – Azerbaijani" lang="az" hreflang="az" data-title="1 + 2 + 3 + 4 + · · ·" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 + 2 + 3 + 4 + · · · – Bosnian" lang="bs" hreflang="bs" data-title="1 + 2 + 3 + 4 + · · ·" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D0%BB%C4%83_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%D1%81%D0%B5%D0%BD_%D1%80%D0%B5%D1%87%C4%95" title="Натураллă хисепсен речĕ – Chuvash" lang="cv" hreflang="cv" data-title="Натураллă хисепсен речĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Summe_aller_nat%C3%BCrlichen_Zahlen" title="Summe aller natürlichen Zahlen – German" lang="de" hreflang="de" data-title="Summe aller natürlichen Zahlen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Spanish" lang="es" hreflang="es" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/..._%2B_%DB%B4_%2B_%DB%B3_%2B_%DB%B2_%2B_%DB%B1" title="... + ۴ + ۳ + ۲ + ۱ – Persian" lang="fa" hreflang="fa" data-title="... + ۴ + ۳ + ۲ + ۱" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – French" lang="fr" hreflang="fr" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A5%A7_%2B_%E0%A5%A8_%2B_%E0%A5%A9_%2B_%E0%A5%AA_%2B_%C2%B7_%C2%B7_%C2%B7" title="१ + २ + ३ + ४ + · · · – Hindi" lang="hi" hreflang="hi" data-title="१ + २ + ३ + ४ + · · ·" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Indonesian" lang="id" hreflang="id" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 + 2 + 3 + 4 + · · · – Italian" lang="it" hreflang="it" data-title="1 + 2 + 3 + 4 + · · ·" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%95%D7%A8_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%98%D7%91%D7%A2%D7%99%D7%99%D7%9D" title="טור המספרים הטבעיים – Hebrew" lang="he" hreflang="he" data-title="טור המספרים הטבעיים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB_%D2%9B%D0%B0%D1%82%D0%B0%D1%80" title="Натурал қатар – Kazakh" lang="kk" hreflang="kk" data-title="Натурал қатар" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/1%2B2%2B3%2B4%2B%E2%80%A6" title="1+2+3+4+… – Japanese" lang="ja" hreflang="ja" data-title="1+2+3+4+…" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Uzbek" lang="uz" hreflang="uz" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pcd mw-list-item"><a href="https://pcd.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Picard" lang="pcd" hreflang="pcd" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Picard" data-language-local-name="Picard" class="interlanguage-link-target"><span>Picard</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szereg_1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%A6" title="Szereg 1 + 2 + 3 + 4 + … – Polish" lang="pl" hreflang="pl" data-title="Szereg 1 + 2 + 3 + 4 + …" data-language-autonym="Polski" data-language-local-name="Polish" 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/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Portuguese" lang="pt" hreflang="pt" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%A6" title="1 + 2 + 3 + 4 + … – Romanian" lang="ro" hreflang="ro" data-title="1 + 2 + 3 + 4 + …" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D1%8F%D0%B4_%D0%B8%D0%B7_%D0%BD%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D1%85_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Ряд из натуральных чисел – Russian" lang="ru" hreflang="ru" data-title="Ряд из натуральных чисел" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7%C2%B7%C2%B7" title="1 + 2 + 3 + 4 + ··· – Slovenian" lang="sl" hreflang="sl" data-title="1 + 2 + 3 + 4 + ···" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Serbian" lang="sr" hreflang="sr" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Thai" lang="th" hreflang="th" data-title="1 + 2 + 3 + 4 + ⋯" 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/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7" title="1 + 2 + 3 + 4 + · · · – Turkish" lang="tr" hreflang="tr" data-title="1 + 2 + 3 + 4 + · · ·" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Ukrainian" lang="uk" hreflang="uk" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯ – Vietnamese" lang="vi" hreflang="vi" data-title="1 + 2 + 3 + 4 + ⋯" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%B8%80%E5%8F%88%E4%BA%8C%E5%8F%88%E4%B8%89%E5%8F%88%E5%9B%9B%E5%8F%88%E2%80%A6" title="一又二又三又四又… – Literary Chinese" lang="lzh" hreflang="lzh" data-title="一又二又三又四又…" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%80%A6" title="1 + 2 + 3 + 4 + … – Chinese" lang="zh" hreflang="zh" data-title="1 + 2 + 3 + 4 + …" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Divergent series</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sum1234Summary.svg" class="mw-file-description"><img alt="A graph depicting the series with layered boxes and a parabola that dips just below the y-axis" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Sum1234Summary.svg/300px-Sum1234Summary.svg.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Sum1234Summary.svg/450px-Sum1234Summary.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Sum1234Summary.svg/600px-Sum1234Summary.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>The first four partial sums of the series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span>. The <a href="/wiki/Parabola" title="Parabola">parabola</a> is their smoothed <a href="/wiki/Asymptote" title="Asymptote">asymptote</a>; its <a href="/wiki/Y-intercept" title="Y-intercept"><i>y</i>-intercept</a> is −1/12.<sup id="cite_ref-Tao_1-0" class="reference"><a href="#cite_note-Tao-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>The infinite series whose terms are the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <b><span class="nowrap">1 + 2 + 3 + 4 + ⋯</span></b> is a <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a>. The <i>n</i>th partial sum of the series is the <a href="/wiki/Triangular_number" title="Triangular number">triangular number</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99476e25466549387c585cb4de44e90f6cbe4cf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.136ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},}"></span></dd></dl> <p>which increases without bound as <i>n</i> goes to <a href="/wiki/Infinity" title="Infinity">infinity</a>. Because the <a href="/wiki/Sequence" title="Sequence">sequence</a> of partial sums fails to <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">converge to a finite limit</a>, the <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a> does not have a sum. </p><p>Although the series seems at first sight not to have any meaningful value at all, it can be manipulated to yield a number of different mathematical results. For example, many <a href="/wiki/Summation_method" class="mw-redirect" title="Summation method">summation methods</a> are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">zeta function regularization</a> and <a href="/wiki/Ramanujan_summation" title="Ramanujan summation">Ramanujan summation</a> assign the series a value of <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>, which is expressed by a famous formula:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1c2edf39416d13e4b29dd0aca3a6270af232499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.449ex; height:5.176ex;" alt="{\displaystyle 1+2+3+4+\cdots =-{\frac {1}{12}},}"></span></dd></dl> <p>where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> in its usual meaning. These methods have applications in other fields such as <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>, and <a href="/wiki/String_theory" title="String theory">string theory</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In a monograph on <a href="/wiki/Moonshine_theory" class="mw-redirect" title="Moonshine theory">moonshine theory</a>, <a href="/wiki/University_of_Alberta" title="University of Alberta">University of Alberta</a> mathematician Terry Gannon calls this equation "one of the most remarkable formulae in science".<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Partial_sums">Partial sums</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=1" title="Edit section: Partial sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:First_six_triangular_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/220px-First_six_triangular_numbers.svg.png" decoding="async" width="220" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/330px-First_six_triangular_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/First_six_triangular_numbers.svg/440px-First_six_triangular_numbers.svg.png 2x" data-file-width="374" data-file-height="314" /></a><figcaption>The first six triangular numbers</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a></div> <p>The partial sums of the series <span class="nowrap">1 + 2 + 3 + 4 + 5 + 6 + ⋯</span> are <span class="nowrap">1, 3, 6, 10, 15</span>, etc. The <i>n</i>th partial sum is given by a simple formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/956e6e212a3c2b7a352b988640a7e67a0dea5727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.136ex; height:6.843ex;" alt="{\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}}.}"></span></dd></dl> <p>This equation was known to the <a href="/wiki/Pythagoreans" class="mw-redirect" title="Pythagoreans">Pythagoreans</a> as early as the sixth century BCE.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Numbers of this form are called <a href="/wiki/Triangular_number" title="Triangular number">triangular numbers</a>, because they can be arranged as an equilateral triangle. </p><p>The infinite sequence of triangular numbers diverges to +∞, so by definition, the infinite series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> also diverges to +∞. The divergence is a simple consequence of the form of the series: the terms do not approach zero, so the series diverges by the <a href="/wiki/Term_test" class="mw-redirect" title="Term test">term test</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Summability">Summability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=2" title="Edit section: Summability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Among the classical divergent series, <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> is relatively difficult to manipulate into a finite value. Many <a href="/wiki/Summation_method" class="mw-redirect" title="Summation method">summation methods</a> are used to assign numerical values to divergent series, some more powerful than others. For example, <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> is a well-known method that sums <a href="/wiki/Grandi%27s_series" title="Grandi's series">Grandi's series</a>, the mildly divergent series <span class="nowrap">1 − 1 + 1 − 1 + ⋯</span>, to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. <a href="/wiki/Abel_summation" class="mw-redirect" title="Abel summation">Abel summation</a> is a more powerful method that not only sums Grandi's series to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, but also sums the trickier series <span class="nowrap"><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></span> to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>. </p><p>Unlike the above series, <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞.<sup id="cite_ref-FOOTNOTEHardy194910_6-0" class="reference"><a href="#cite_note-FOOTNOTEHardy194910-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum <span class="nowrap">1 + 2 + 3 + ⋯</span> to a finite value <style data-mw-deduplicate="TemplateStyles:r1033199720">.mw-parser-output div.crossreference{padding-left:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><span role="note" class="hatnote navigation-not-searchable crossreference">(see <i><a href="#Heuristics">§ Heuristics</a></i> below)</span>. More advanced methods are required, such as <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">zeta function regularization</a> or <a href="/wiki/Ramanujan_summation" title="Ramanujan summation">Ramanujan summation</a>. It is also possible to argue for the value of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> using some rough heuristics related to these methods. </p> <div class="mw-heading mw-heading3"><h3 id="Heuristics">Heuristics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=3" title="Edit section: Heuristics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/49/Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg/400px-Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg" decoding="async" width="400" height="107" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/49/Ramanujan_Notebook_1_Chapter_8_on_1234_series.jpg 1.5x" data-file-width="450" data-file-height="120" /></a><figcaption>Passage from <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan</a>'s first notebook describing the "constant" of the series</figcaption></figure> <p><a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan</a> presented two derivations of "<span class="nowrap">1 + 2 + 3 + 4 + ⋯ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></span>" in chapter 8 of his first notebook.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The simpler, less rigorous derivation proceeds in two steps, as follows. </p><p>The first key insight is that the series of positive numbers <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> closely resembles the <a href="/wiki/Alternating_series" title="Alternating series">alternating series</a> <span class="nowrap"><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></span>. The latter series is also divergent, but it is much easier to work with; there are several classical methods that assign it a value, which have been explored since the 18th century.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In order to transform the series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> into <span class="nowrap">1 − 2 + 3 − 4 + ⋯</span>, one can subtract 4 from the second term, 8 from the fourth term, 12 from the sixth term, and so on. The total amount to be subtracted is <span class="nowrap">4 + 8 + 12 + 16 + ⋯</span>, which is 4 times the original series. These relationships can be expressed using algebra. Whatever the "sum" of the series might be, call it <span class="nowrap"><i>c</i> = 1 + 2 + 3 + 4 + ⋯.</span> Then multiply this equation by 4 and subtract the second equation from the first: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{7}c={}&&1+2&&{}+3+4&&{}+5+6+\cdots \\4c={}&&4&&{}+8&&{}+12+\cdots \\c-4c={}&&1-2&&{}+3-4&&{}+5-6+\cdots \end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd /> <mtd> <mn>1</mn> <mo>+</mo> <mn>2</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>5</mn> <mo>+</mo> <mn>6</mn> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd /> <mtd> <mn>4</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>8</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>12</mn> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>−</mo> <mn>4</mn> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd /> <mtd> <mn>1</mn> <mo>−</mo> <mn>2</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>3</mn> <mo>−</mo> <mn>4</mn> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>5</mn> <mo>−</mo> <mn>6</mn> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}c={}&&1+2&&{}+3+4&&{}+5+6+\cdots \\4c={}&&4&&{}+8&&{}+12+\cdots \\c-4c={}&&1-2&&{}+3-4&&{}+5-6+\cdots \end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e5affe1303c16802fd88806fc793bdb09524a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:36.607ex; height:8.843ex;" alt="{\displaystyle {\begin{alignedat}{7}c={}&&1+2&&{}+3+4&&{}+5+6+\cdots \\4c={}&&4&&{}+8&&{}+12+\cdots \\c-4c={}&&1-2&&{}+3-4&&{}+5-6+\cdots \end{alignedat}}}"></span></dd></dl> <p>The second key insight is that the alternating series <span class="nowrap">1 − 2 + 3 − 4 + ⋯</span> is the formal <a href="/wiki/Power_series" title="Power series">power series</a> expansion (for <i>x</i> at point 0) of the function <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">(1 + <i>x</i>)<sup>2</sup></span></span>⁠</span> which is <span class="nowrap">1 − 2x + 3x^2 − 4x^3 + ⋯</span> evaluated with <i>x</i> defined as 1. Accordingly, Ramanujan writes </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 -3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>c</mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/febbbe1006527c5029d346860a464df2462af285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.518ex; height:6.009ex;" alt="{\displaystyle -3c=1-2+3-4+\cdots ={\frac {1}{(1+1)^{2}}}={\frac {1}{4}}.}"></span></dd></dl> <p>Dividing both sides by −3, one gets <i>c</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>. </p><p>Generally speaking, it is incorrect to manipulate infinite series as if they were finite sums. For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step <span class="nowrap">4<i>c</i> = 0 + 4 + 0 + 8 + ⋯</span> is not justified by the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a> law alone. For an extreme example, appending a single zero to the front of the series can lead to a different result.<sup id="cite_ref-Tao_1-1" class="reference"><a href="#cite_note-Tao-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>One way to remedy this situation, and to constrain the places where zeroes may be inserted, is to keep track of each term in the series by attaching a dependence on some function.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In the series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span>, each term <i>n</i> is just a number. If the term <i>n</i> is promoted to a function <i>n</i><sup><i>−s</i></sup>, where <i>s</i> is a complex variable, then one can ensure that only like terms are added. The resulting series may be manipulated in a more rigorous fashion, and the variable <i>s</i> can be set to −1 later. The implementation of this strategy is called <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">zeta function regularization</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Zeta_function_regularization">Zeta function regularization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=4" title="Edit section: Zeta function regularization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Zeta_plot.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Zeta_plot.gif/310px-Zeta_plot.gif" decoding="async" width="310" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Zeta_plot.gif/465px-Zeta_plot.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Zeta_plot.gif/620px-Zeta_plot.gif 2x" data-file-width="818" data-file-height="528" /></a><figcaption>Plot of <i>ζ</i>(<i>s</i>). For <span class="nowrap"><i>s</i> > 1</span>, the series converges and <span class="nowrap"><i>ζ</i>(<i>s</i>) > 1</span>. Analytic continuation around the pole at <span class="nowrap"><i>s</i> = 1</span> leads to a region of negative values, including <span class="nowrap"><i>ζ</i>(−1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></span>.</figcaption></figure> <p>In <a href="/wiki/Zeta_function_regularization" title="Zeta function regularization">zeta function regularization</a>, the series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f1f1df68f6fa171643c75e2b2fcd756eb3f547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.555ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }n}"></span> is replaced by the series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{n=1}^{\infty }n^{-s}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{n=1}^{\infty }n^{-s}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38748e061a71cb7b07f7b0bbe073964ba8d3884a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.483ex; height:3.176ex;" alt="{\textstyle \sum _{n=1}^{\infty }n^{-s}.}"></span> The latter series is an example of a <a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a>. When the real part of <i>s</i> is greater than 1, the Dirichlet series converges, and its sum is the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> <i>ζ</i>(<i>s</i>). On the other hand, the Dirichlet series diverges when the real part of <i>s</i> is less than or equal to 1, so, in particular, the series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> that results from setting <span class="nowrap"><i>s</i> = −1</span> does not converge. The benefit of introducing the Riemann zeta function is that it can be defined for other values of <i>s</i> by <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>. One can then define the zeta-regularized sum of <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> to be <i>ζ</i>(−1). </p><p>From this point, there are a few ways to prove that <span class="nowrap"><i>ζ</i>(−1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>.</span> One method, along the lines of Euler's reasoning,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> uses the relationship between the Riemann zeta function and the <a href="/wiki/Dirichlet_eta_function" title="Dirichlet eta function">Dirichlet eta function</a> <i>η</i>(<i>s</i>). The eta function is defined by an alternating Dirichlet series, so this method parallels the earlier heuristics. Where both Dirichlet series converge, one has the identities: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s).\end{alignedat}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em 0em" displaystyle="true"> <mtr> <mtd> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mn>2</mn> <mo>×</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mn>2</mn> <mo>×</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>−</mo> <msup> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s).\end{alignedat}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a20c0748bee27ad253ac8c15d160ab9238a7cc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:66.877ex; height:9.843ex;" alt="{\displaystyle {\begin{alignedat}{7}\zeta (s)&{}={}&1^{-s}+2^{-s}&&{}+3^{-s}+4^{-s}&&{}+5^{-s}+6^{-s}+\cdots &\\2\times 2^{-s}\zeta (s)&{}={}&2\times 2^{-s}&&{}+2\times 4^{-s}&&{}+2\times 6^{-s}+\cdots &\\\left(1-2^{1-s}\right)\zeta (s)&{}={}&1^{-s}-2^{-s}&&{}+3^{-s}-4^{-s}&&{}+5^{-s}-6^{-s}+\cdots &=\eta (s).\end{alignedat}}}"></span></dd></dl> <p>The identity <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 (1-2^{1-s})\zeta (s)=\eta (s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>s</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-2^{1-s})\zeta (s)=\eta (s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4eb911624d01c0d5edf51dd1a844c82d6c98084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.241ex; height:3.176ex;" alt="{\displaystyle (1-2^{1-s})\zeta (s)=\eta (s)}"></span> continues to hold when both functions are extended by analytic continuation to include values of <i>s</i> for which the above series diverge. Substituting <span class="nowrap"><i>s</i> = −1</span>, one gets <span class="nowrap">−3<i>ζ</i>(−1) = <i>η</i>(−1)</span>. Now, computing <i>η</i>(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> which is a <a href="/wiki/One-sided_limit" title="One-sided limit">one-sided limit</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>ζ</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo stretchy="false">→</mo> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f194f6afbd248171818712f5403b210825cfd46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:75.818ex; height:6.009ex;" alt="{\displaystyle -3\zeta (-1)=\eta (-1)=\lim _{x\to 1^{-}}\left(1-2x+3x^{2}-4x^{3}+\cdots \right)=\lim _{x\to 1^{-}}{\frac {1}{(1+x)^{2}}}={\frac {1}{4}}.}"></span></dd></dl> <p>Dividing both sides by −3, one gets <span class="nowrap"><i>ζ</i>(−1) = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>.</span> </p> <div class="mw-heading mw-heading3"><h3 id="Cutoff_regularization">Cutoff regularization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=5" title="Edit section: Cutoff regularization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:408px;max-width:408px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Sum1234Plain.svg" class="mw-file-description"><img alt="A graph depicting the series with layered boxes" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Sum1234Plain.svg/200px-Sum1234Plain.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Sum1234Plain.svg/300px-Sum1234Plain.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Sum1234Plain.svg/400px-Sum1234Plain.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div><div class="thumbcaption">The series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span></div></div><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Sum1234Smoothed.svg" class="mw-file-description"><img alt="A graph depicting the smoothed series with layered curving stripes" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Sum1234Smoothed.svg/200px-Sum1234Smoothed.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Sum1234Smoothed.svg/300px-Sum1234Smoothed.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Sum1234Smoothed.svg/400px-Sum1234Smoothed.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></div><div class="thumbcaption">After smoothing</div></div></div></div></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sum1234Asymptote.svg" class="mw-file-description"><img alt="A graph showing a parabola that dips just below the y-axis" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Sum1234Asymptote.svg/220px-Sum1234Asymptote.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Sum1234Asymptote.svg/330px-Sum1234Asymptote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Sum1234Asymptote.svg/440px-Sum1234Asymptote.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Asymptotic behavior of the smoothing. The <i>y</i>-intercept of the parabola is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>.<sup id="cite_ref-Tao_1-2" class="reference"><a href="#cite_note-Tao-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <p>The method of regularization using a <a href="/wiki/Cutoff_function" class="mw-redirect" title="Cutoff function">cutoff function</a> can "smooth" the series to arrive at <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>. Smoothing is a conceptual bridge between zeta function regularization, with its reliance on <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, and Ramanujan summation, with its shortcut to the <a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>. Instead, the method operates directly on conservative transformations of the series, using methods from <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>. </p><p>The idea is to replace the ill-behaved discrete series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{n=0}^{N}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>n</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=0}^{N}n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ebfbf633842ad6ba0a19574c5861095129bd67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.555ex; height:3.509ex;" alt="{\displaystyle \textstyle \sum _{n=0}^{N}n}"></span> with a smoothed version </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 _{n=0}^{\infty }nf\left({\frac {n}{N}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <mi>n</mi> <mi>f</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mi>N</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }nf\left({\frac {n}{N}}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162774159c5c23c4861327eb48ce5f15525cd193" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:13.512ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }nf\left({\frac {n}{N}}\right),}"></span></dd></dl> <p>where <i>f</i> is a cutoff function with appropriate properties. The cutoff function must be normalized to <span class="nowrap"><i>f</i>(0) = 1</span>; this is a different normalization from the one used in differential equations. The cutoff function should have enough bounded derivatives to smooth out the wrinkles in the series, and it should decay to 0 faster than the series grows. For convenience, one may require that <i>f</i> is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a>, <a href="/wiki/Bounded_function" title="Bounded function">bounded</a>, and <a href="/wiki/Compactly_supported" class="mw-redirect" title="Compactly supported">compactly supported</a>. One can then prove that this smoothed sum is <a href="/wiki/Asymptotic" class="mw-redirect" title="Asymptotic">asymptotic</a> to <span class="nowrap"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> + <i>CN</i><sup>2</sup></span>, where <i>C</i> is a constant that depends on <i>f</i>. The constant term of the asymptotic expansion does not depend on <i>f</i>: it is necessarily the same value given by analytic continuation, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>.<sup id="cite_ref-Tao_1-3" class="reference"><a href="#cite_note-Tao-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ramanujan_summation">Ramanujan summation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=6" title="Edit section: Ramanujan summation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Ramanujan_summation" title="Ramanujan summation">Ramanujan sum</a> of <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> is also <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>. Ramanujan wrote in his second letter to <a href="/wiki/G._H._Hardy" title="G. H. Hardy">G. H. Hardy</a>, dated 27 February 1913: </p> <dl><dd>"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully <a href="/wiki/Thomas_John_I%27Anson_Bromwich" title="Thomas John I'Anson Bromwich">Bromwich</a>'s <i>Infinite Series</i> and not fall into the pitfalls of divergent series. ... I told him that the sum of an infinite number of terms of the series: <span class="nowrap">1 + 2 + 3 + 4 + ⋯ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></span> under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. ..."<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Ramanujan summation is a method to isolate the constant term in the <a href="/wiki/Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a> for the partial sums of a series. For a function <i>f</i>, the classical Ramanujan sum of the series <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6577fdc4735fd8c269a5350ca8f08b41a92bebd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.329ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{k=1}^{\infty }f(k)}"></span> is defined as </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 c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>k</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b4d5bcd4404319b06ccf0315e6947c05ba5dd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.64ex; height:6.843ex;" alt="{\displaystyle c=-{\frac {1}{2}}f(0)-\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(0),}"></span></dd></dl> <p>where <i>f</i><sup>(2<i>k</i>−1)</sup> is the (2<i>k</i> − 1)th derivative of <i>f</i> and <i>B</i><sub>2<i>k</i></sub> is the (2<i>k</i>)th <a href="/wiki/Bernoulli_number" title="Bernoulli number">Bernoulli number</a>: <span class="nowrap"><i>B</i><sub>2</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span></span>, <span class="nowrap"><i>B</i><sub>4</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">30</span></span>⁠</span></span>, and so on. Setting <span class="nowrap"><i>f</i>(<i>x</i>) = <i>x</i></span>, the first derivative of <i>f</i> is 1, and every other term vanishes, so<sup id="cite_ref-Berndt_1985_13,134_15-0" class="reference"><a href="#cite_note-Berndt_1985_13,134-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ca43136f0173869e29d219b8d455884d8805f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.112ex; height:5.343ex;" alt="{\displaystyle c=-{\frac {1}{6}}\times {\frac {1}{2!}}=-{\frac {1}{12}}.}"></span></dd></dl> <p>To avoid inconsistencies, the modern theory of Ramanujan summation requires that <i>f</i> is "regular" in the sense that the higher-order derivatives of <i>f</i> decay quickly enough for the remainder terms in the Euler–Maclaurin formula to tend to 0. Ramanujan tacitly assumed this property.<sup id="cite_ref-Berndt_1985_13,134_15-1" class="reference"><a href="#cite_note-Berndt_1985_13,134-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The regularity requirement prevents the use of Ramanujan summation upon spaced-out series like <span class="nowrap">0 + 2 + 0 + 4 + ⋯</span>, because no regular function takes those values. Instead, such a series must be interpreted by zeta function regularization. For this reason, Hardy recommends "great caution" when applying the Ramanujan sums of known series to find the sums of related series.<sup id="cite_ref-FOOTNOTEHardy1949346_16-0" class="reference"><a href="#cite_note-FOOTNOTEHardy1949346-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Failure_of_stable_linear_summation_methods">Failure of stable linear summation methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=7" title="Edit section: Failure of stable linear summation methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></div> <p>A summation method that is <a href="/wiki/Divergent_series#Properties_of_summation_methods" title="Divergent series">linear and stable</a> cannot sum the series <span class="texhtml">1 + 2 + 3 + ⋯</span> to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If </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+2+3+\cdots =x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2+3+\cdots =x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70cdf51bfb444d2656ddf3cb568ff2d68037017" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.807ex; height:2.509ex;" alt="{\displaystyle 1+2+3+\cdots =x,}"></span></dd></dl> <p>then adding 0 to both sides gives </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 0+1+2+3+\cdots =0+x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>+</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+1+2+3+\cdots =0+x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4318027185f56073d8ef61e656b3c4039c21796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:31.594ex; height:2.343ex;" alt="{\displaystyle 0+1+2+3+\cdots =0+x=x}"></span></dd></dl> <p>by stability. By linearity, one may subtract the second equation from the first (subtracting each component of the second line from the first line in columns) to give </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+1+1+\cdots =x-x=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+1+\cdots =x-x=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aabd193260dd8bd5f0f1a15132f39c3adcdede0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.238ex; height:2.343ex;" alt="{\displaystyle 1+1+1+\cdots =x-x=0.}"></span></dd></dl> <p>Adding 0 to both sides again gives </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 0+1+1+1+\cdots =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+1+1+1+\cdots =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93094ba09d21865ab71814a8df5fa2b79baac39b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.642ex; height:2.509ex;" alt="{\displaystyle 0+1+1+1+\cdots =0,}"></span></dd></dl> <p>and subtracting the last two series gives </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+0+0+0+\cdots =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mn>0</mn> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+0+0+0+\cdots =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47b2a3680339fb1f184f38435661ca52110cf2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.642ex; height:2.509ex;" alt="{\displaystyle 1+0+0+0+\cdots =0,}"></span></dd></dl> <p>contradicting stability. </p><p>Therefore, every method that gives a finite value to the sum <span class="nowrap">1 + 2 + 3 + ⋯</span> is not stable or not linear.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Physics">Physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=8" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">bosonic string theory</a>, the attempt is to compute the possible energy levels of a string, in particular, the lowest energy level. Speaking informally, each harmonic of the string can be viewed as a collection of <span class="nowrap"><i>D</i> − 2</span> independent <a href="/wiki/Quantum_harmonic_oscillator" title="Quantum harmonic oscillator">quantum harmonic oscillators</a>, one for each <a href="/wiki/Transverse_wave" title="Transverse wave">transverse direction</a>, where <i>D</i> is the dimension of spacetime. If the fundamental oscillation frequency is <i>ω</i>, then the energy in an oscillator contributing to the <i>n</i>th harmonic is <i>nħω</i>/2. So using the divergent series, the sum over all harmonics is <span class="nowrap">−<i>ħω</i>(<i>D</i> − 2)/24</span>. Ultimately it is this fact, combined with the <a href="/wiki/Goddard%E2%80%93Thorn_theorem" title="Goddard–Thorn theorem">Goddard–Thorn theorem</a>, which leads to bosonic string theory failing to be consistent in dimensions other than 26.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>The regularization of <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> is also involved in computing the <a href="/wiki/Casimir_force#Derivation_of_Casimir_effect_assuming_zeta-regularization" class="mw-redirect" title="Casimir force">Casimir force</a> for a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> in one dimension.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> An exponential cutoff function suffices to smooth the series, representing the fact that arbitrarily high-energy modes are not blocked by the conducting plates. The spatial symmetry of the problem is responsible for canceling the quadratic term of the expansion. All that is left is the constant term −1/12, and the negative sign of this result reflects the fact that the Casimir force is attractive.<sup id="cite_ref-FOOTNOTEZee2003pp._65–67_20-0" class="reference"><a href="#cite_note-FOOTNOTEZee2003pp._65–67-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>A similar calculation is involved in three dimensions, using the <a href="/wiki/Real_analytic_Eisenstein_series#Epstein_zeta_function" title="Real analytic Eisenstein series">Epstein zeta-function</a> in place of the Riemann zeta function.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=9" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is unclear whether Leonhard Euler summed the series to <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span>. According to <a href="/wiki/Morris_Kline" title="Morris Kline">Morris Kline</a>, Euler's early work on divergent series relied on function expansions, from which he concluded <span class="nowrap">1 + 2 + 3 + 4 + ⋯ = ∞</span>.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> According to Raymond Ayoub, the fact that the divergent zeta series is not Abel-summable prevented Euler from using the zeta function as freely as the eta function, and he "could not have attached a meaning" to the series.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Other authors have credited Euler with the sum, suggesting that Euler would have extended the relationship between the zeta and eta functions to negative integers.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In the primary literature, the series <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> is mentioned in Euler's 1760 publication <span title="Latin-language text"><i lang="la">De seriebus divergentibus</i></span> alongside the divergent geometric series <span class="nowrap"><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></span>. Euler hints that series of this type have finite, negative sums, and he explains what this means for geometric series, but he does not return to discuss <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span>. In the same publication, Euler writes that the sum of <span class="nowrap"><a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></span> is infinite.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_popular_media">In popular media</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=10" title="Edit section: In popular media"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/David_Leavitt" title="David Leavitt">David Leavitt</a>'s 2007 novel <i><a href="/wiki/The_Indian_Clerk" title="The Indian Clerk">The Indian Clerk</a></i> includes a scene where Hardy and <a href="/wiki/John_Edensor_Littlewood" title="John Edensor Littlewood">Littlewood</a> discuss the meaning of this series. They conclude that Ramanujan has rediscovered <i>ζ</i>(−1), and they take the "lunatic asylum" line in his second letter as a sign that Ramanujan is toying with them.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Simon_McBurney" title="Simon McBurney">Simon McBurney</a>'s 2007 play <i><a href="/wiki/A_Disappearing_Number" title="A Disappearing Number">A Disappearing Number</a></i> focuses on the series in the opening scene. The main character, Ruth, walks into a lecture hall and introduces the idea of a divergent series before proclaiming, "I'm going to show you something really thrilling", namely <span class="nowrap">1 + 2 + 3 + 4 + ⋯ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></span>. As Ruth launches into a derivation of the functional equation of the zeta function, another actor addresses the audience, admitting that they are actors: "But the mathematics is real. It's terrifying, but it's real."<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p>In January 2014, <a href="/wiki/Numberphile" title="Numberphile">Numberphile</a> produced a <a href="/wiki/YouTube" title="YouTube">YouTube</a> video on the series, which gathered over 1.5 million views in its first month.<sup id="cite_ref-NYT-20140203_31-0" class="reference"><a href="#cite_note-NYT-20140203-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> The 8-minute video is narrated by Tony Padilla, a physicist at the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>. Padilla begins with <span class="nowrap">1 − 1 + 1 − 1 + ⋯</span> and <span class="nowrap">1 − 2 + 3 − 4 + ⋯</span> and relates the latter to <span class="nowrap">1 + 2 + 3 + 4 + ⋯</span> using a term-by-term subtraction similar to Ramanujan's argument.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Numberphile also released a 21-minute version of the video featuring Nottingham physicist <a href="/wiki/Edmund_Copeland" title="Edmund Copeland">Ed Copeland</a>, who describes in more detail how <span class="nowrap">1 − 2 + 3 − 4 + ⋯ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></span> as an Abel sum, and <span class="nowrap">1 + 2 + 3 + 4 + ⋯ = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span></span> as <i>ζ</i>(−1).<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> After receiving complaints about the lack of rigour in the first video, Padilla also wrote an explanation on his webpage relating the manipulations in the video to identities between the analytic continuations of the relevant Dirichlet series.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>In <i><a href="/wiki/The_New_York_Times" title="The New York Times">The New York Times</a></i> coverage of the Numberphile video, mathematician <a href="/wiki/Edward_Frenkel" title="Edward Frenkel">Edward Frenkel</a> commented: "This calculation is one of the best-kept secrets in math. No one on the outside knows about it."<sup id="cite_ref-NYT-20140203_31-1" class="reference"><a href="#cite_note-NYT-20140203-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Coverage of this topic in <a href="/wiki/Smithsonian_(magazine)" title="Smithsonian (magazine)"><i>Smithsonian</i> magazine</a> describes the Numberphile video as misleading and notes that the interpretation of the sum as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">12</span></span>⁠</span> relies on a specialized meaning for the <i>equals</i> sign, from the techniques of <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a>, in which <i>equals</i> means <i>is associated with</i>.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The Numberphile video was critiqued on similar grounds by German mathematician <a href="/wiki/Burkard_Polster" title="Burkard Polster">Burkard Polster</a> on his <i>Mathologer</i> YouTube channel in 2018, his video receiving 2.7 million views by 2023.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Tao-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Tao_1-0"><sup><i><b>a</b></i></sup></a> <a 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a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output 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National Center for Mathematics Education, University of Gothenburg, Sweden. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-9185143009" title="Special:BookSources/978-9185143009"><bdi>978-9185143009</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+bridge+between+the+continuous+and+the+discrete+via+original+sources&rft.btitle=Study+the+Masters%3A+The+Abel-Fauvel+Conference&rft.pages=3&rft.pub=National+Center+for+Mathematics+Education%2C+University+of+Gothenburg%2C+Sweden&rft.date=2002&rft.isbn=978-9185143009&rft.aulast=Pengelley&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-FOOTNOTEHardy194910-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHardy194910_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardy1949">Hardy 1949</a>, p. 10.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook1/chapterVIII/page3.htm"><i>Ramanujan's Notebooks</i></a><span class="reference-accessdate">, retrieved <span class="nowrap">January 26,</span> 2014</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks&rft_id=http%3A%2F%2Fwww.imsc.res.in%2F~rao%2Framanujan%2FNoteBooks%2FNoteBook1%2FchapterVIII%2Fpage3.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbdi1992" class="citation cs2">Abdi, Wazir Hasan (1992), <i>Toils and triumphs of Srinivasa Ramanujan, the man and the mathematician</i>, National, p. 41</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Toils+and+triumphs+of+Srinivasa+Ramanujan%2C+the+man+and+the+mathematician&rft.pages=41&rft.pub=National&rft.date=1992&rft.aulast=Abdi&rft.aufirst=Wazir+Hasan&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1985" class="citation cs2">Berndt, Bruce C. (1985), <i>Ramanujan's Notebooks: Part 1</i>, Springer-Verlag, pp. 135–136</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+1&rft.pages=135-136&rft.pub=Springer-Verlag&rft.date=1985&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler2006" class="citation web cs1">Euler, Leonhard (2006). <a rel="nofollow" class="external text" href="http://www.math.dartmouth.edu/~euler/pages/E352.html">"Translation with notes of Euler's paper: Remarks on a beautiful relation between direct as well as reciprocal power series"</a>. Translated by Willis, Lucas; Osler, Thomas J. The Euler Archive<span class="reference-accessdate">. Retrieved <span class="nowrap">2007-03-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Translation+with+notes+of+Euler%27s+paper%3A+Remarks+on+a+beautiful+relation+between+direct+as+well+as+reciprocal+power+series&rft.pub=The+Euler+Archive&rft.date=2006&rft.aulast=Euler&rft.aufirst=Leonhard&rft_id=http%3A%2F%2Fwww.math.dartmouth.edu%2F~euler%2Fpages%2FE352.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span> Originally published as <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEuler1768" class="citation journal cs1 cs1-prop-foreign-lang-source">Euler, Leonhard (1768). "Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques". <i>Mémoires de l'Académie des Sciences de Berlin</i> (in French). <b>17</b>: 83–106.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+des+Sciences+de+Berlin&rft.atitle=Remarques+sur+un+beau+rapport+entre+les+s%C3%A9ries+des+puissances+tant+directes+que+r%C3%A9ciproques&rft.volume=17&rft.pages=83-106&rft.date=1768&rft.aulast=Euler&rft.aufirst=Leonhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Promoting numbers to functions is identified as one of two broad classes of summation methods, including Abel and Borel summation, by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnopp1990" class="citation book cs1"><a href="/wiki/Konrad_Knopp" title="Konrad Knopp">Knopp, Konrad</a> (1990) [1922]. <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00knop_448"><i>Theory and Application of Infinite Series</i></a></span>. Dover. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00knop_448/page/n487">475</a>–476. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66165-2" title="Special:BookSources/0-486-66165-2"><bdi>0-486-66165-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Application+of+Infinite+Series&rft.pages=475-476&rft.pub=Dover&rft.date=1990&rft.isbn=0-486-66165-2&rft.aulast=Knopp&rft.aufirst=Konrad&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryapplicatio00knop_448&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStopple2003" class="citation cs2">Stopple, Jeffrey (2003), <i>A Primer of Analytic Number Theory: From Pythagoras to Riemann</i>, p. 202, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-81309-3" title="Special:BookSources/0-521-81309-3"><bdi>0-521-81309-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Primer+of+Analytic+Number+Theory%3A+From+Pythagoras+to+Riemann&rft.pages=202&rft.date=2003&rft.isbn=0-521-81309-3&rft.aulast=Stopple&rft.aufirst=Jeffrey&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnopp1990" class="citation book cs1"><a href="/wiki/Konrad_Knopp" title="Konrad Knopp">Knopp, Konrad</a> (1990) [1922]. <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00knop_448"><i>Theory and Application of Infinite Series</i></a></span>. Dover. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/theoryapplicatio00knop_448/page/n502">490</a>–492. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-66165-2" title="Special:BookSources/0-486-66165-2"><bdi>0-486-66165-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+and+Application+of+Infinite+Series&rft.pages=490-492&rft.pub=Dover&rft.date=1990&rft.isbn=0-486-66165-2&rft.aulast=Knopp&rft.aufirst=Konrad&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryapplicatio00knop_448&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAiyangar1995" class="citation book cs1">Aiyangar, Srinivasa Ramanujan (7 September 1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Of5G0r6DQiEC&pg=PA53"><i>Ramanujan: Letters and Commentary</i></a>. p. 53. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821891254" title="Special:BookSources/9780821891254"><bdi>9780821891254</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+Letters+and+Commentary&rft.pages=53&rft.date=1995-09-07&rft.isbn=9780821891254&rft.aulast=Aiyangar&rft.aufirst=Srinivasa+Ramanujan&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOf5G0r6DQiEC%26pg%3DPA53&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> <li id="cite_note-Berndt_1985_13,134-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-Berndt_1985_13,134_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Berndt_1985_13,134_15-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndt1985" class="citation cs2">Berndt, Bruce C. (1985), <i>Ramanujan's Notebooks: Part 1</i>, Springer-Verlag, pp. 13, 134</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%27s+Notebooks%3A+Part+1&rft.pages=13%2C+134&rft.pub=Springer-Verlag&rft.date=1985&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-FOOTNOTEHardy1949346-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHardy1949346_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardy1949">Hardy 1949</a>, p. 346.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNatielloSolari2015" class="citation cs2">Natiello, Mario A.; Solari, Hernan Gustavo (July 2015), "On the removal of infinities from divergent series", <i><a href="/wiki/Philosophy_of_Mathematics_Education_Journal" title="Philosophy of Mathematics Education Journal">Philosophy of Mathematics Education Journal</a></i>, <b>29</b>: 1–11, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11336%2F46148">11336/46148</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophy+of+Mathematics+Education+Journal&rft.atitle=On+the+removal+of+infinities+from+divergent+series&rft.volume=29&rft.pages=1-11&rft.date=2015-07&rft_id=info%3Ahdl%2F11336%2F46148&rft.aulast=Natiello&rft.aufirst=Mario+A.&rft.au=Solari%2C+Hernan+Gustavo&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarbiellini1987" class="citation cs2">Barbiellini, Bernardo (1987), "The Casimir effect in conformal field theories", <i>Physics Letters B</i>, <b>190</b> (1–2): 137–139, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1987PhLB..190..137B">1987PhLB..190..137B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0370-2693%2887%2990854-9">10.1016/0370-2693(87)90854-9</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=The+Casimir+effect+in+conformal+field+theories&rft.volume=190&rft.issue=1%E2%80%932&rft.pages=137-139&rft.date=1987&rft_id=info%3Adoi%2F10.1016%2F0370-2693%2887%2990854-9&rft_id=info%3Abibcode%2F1987PhLB..190..137B&rft.aulast=Barbiellini&rft.aufirst=Bernardo&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">See <a href="https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension" class="extiw" title="v:Quantum mechanics/Casimir effect in one dimension">v:Quantum mechanics/Casimir effect in one dimension</a>.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources"><span title="The material near this tag may rely on an unreliable source. (May 2021)">unreliable source?</span></a></i>]</sup></span> </li> <li id="cite_note-FOOTNOTEZee2003pp._65–67-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEZee2003pp._65–67_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFZee2003">Zee 2003</a>, pp. 65–67.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZeidler2007" class="citation cs2">Zeidler, Eberhard (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XYtnGl9enNgC&pg=PA305">"Quantum Field Theory I: Basics in Mathematics and Physics: A Bridge between Mathematicians and Physicists"</a>, <i>Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists</i>, Springer: 305–306, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006qftb.book.....Z">2006qftb.book.....Z</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540347644" title="Special:BookSources/9783540347644"><bdi>9783540347644</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quantum+Field+Theory+I%3A+Basics+in+Mathematics+and+Physics.+A+Bridge+Between+Mathematicians+and+Physicists&rft.atitle=Quantum+Field+Theory+I%3A+Basics+in+Mathematics+and+Physics%3A+A+Bridge+between+Mathematicians+and+Physicists&rft.pages=305-306&rft.date=2007&rft_id=info%3Abibcode%2F2006qftb.book.....Z&rft.isbn=9783540347644&rft.aulast=Zeidler&rft.aufirst=Eberhard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXYtnGl9enNgC%26pg%3DPA305&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span>.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1983" class="citation cs2"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (November 1983), "Euler and Infinite Series", <i>Mathematics Magazine</i>, <b>56</b> (5): 307–314, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.639.6923">10.1.1.639.6923</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690371">10.2307/2690371</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690371">2690371</a></cite><span 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Retrieved <span class="nowrap">August 31,</span> 2023</span> – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Numberphile+v.+Math%3A+the+truth+about+1%2B2%2B3%2B...%3D-1%2F12&rft.date=2018-01-13&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DYuIIjLr6vUA&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=12" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerndtSrinivasa_Ramanujan_AiyangarRankin1995" class="citation book cs1">Berndt, Bruce C.; <a href="/wiki/Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Srinivasa Ramanujan Aiyangar</a>; Rankin, Robert A. (1995). <i>Ramanujan: letters and commentary</i>. American Mathematical Society. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0287-9" title="Special:BookSources/0-8218-0287-9"><bdi>0-8218-0287-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ramanujan%3A+letters+and+commentary&rft.pub=American+Mathematical+Society&rft.date=1995&rft.isbn=0-8218-0287-9&rft.aulast=Berndt&rft.aufirst=Bruce+C.&rft.au=Srinivasa+Ramanujan+Aiyangar&rft.au=Rankin%2C+Robert+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardy1949" class="citation book cs1"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a> (1949). <i>Divergent Series</i>. Clarendon Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Divergent+Series&rft.pub=Clarendon+Press&rft.date=1949&rft.aulast=Hardy&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZee2003" class="citation book cs1">Zee, A. (2003). <i>Quantum field theory in a nutshell</i>. Princeton UP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-01019-6" title="Special:BookSources/0-691-01019-6"><bdi>0-691-01019-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+field+theory+in+a+nutshell&rft.pub=Princeton+UP&rft.date=2003&rft.isbn=0-691-01019-6&rft.aulast=Zee&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=13" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZwiebach2004" class="citation book cs1">Zwiebach, Barton (2004). <i>A First Course in String Theory</i>. Cambridge UP. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-83143-1" title="Special:BookSources/0-521-83143-1"><bdi>0-521-83143-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+String+Theory&rft.pub=Cambridge+UP&rft.date=2004&rft.isbn=0-521-83143-1&rft.aulast=Zwiebach&rft.aufirst=Barton&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span> See p. 293.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElizalde2004" class="citation encyclopaedia cs1">Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". <i>Proceedings of the II International Conference on Fundamental Interactions</i>. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/gr-qc/0409076">gr-qc/0409076</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004gr.qc.....9076E">2004gr.qc.....9076E</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Cosmology%3A+Techniques+and+Applications&rft.btitle=Proceedings+of+the+II+International+Conference+on+Fundamental+Interactions&rft.date=2004&rft_id=info%3Aarxiv%2Fgr-qc%2F0409076&rft_id=info%3Abibcode%2F2004gr.qc.....9076E&rft.aulast=Elizalde&rft.aufirst=Emilio&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWatson1929" class="citation cs2">Watson, G. N. (April 1929), "Theorems stated by Ramanujan (VIII): Theorems on Divergent Series", <i>Journal of the London Mathematical Society</i>, 1, <b>4</b> (2): 82–86, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-4.14.82">10.1112/jlms/s1-4.14.82</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+London+Mathematical+Society&rft.atitle=Theorems+stated+by+Ramanujan+%28VIII%29%3A+Theorems+on+Divergent+Series&rft.volume=4&rft.issue=2&rft.pages=82-86&rft.date=1929-04&rft_id=info%3Adoi%2F10.1112%2Fjlms%2Fs1-4.14.82&rft.aulast=Watson&rft.aufirst=G.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png" decoding="async" width="40" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/60px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/80px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Wikiversity has learning resources about <i><b><a href="https://en.wikiversity.org/wiki/divergent_series" class="extiw" title="v:divergent series">divergent series</a></b></i></div></div> </div> <ul><li>Lamb E. (2014), "<a rel="nofollow" class="external text" href="http://blogs.scientificamerican.com/roots-of-unity/2014/01/20/is-the-sum-of-positive-integers-negative/">Does 1+2+3... Really Equal –1/12?</a>", <a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a> Blogs.</li> <li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week124.html">This Week's Finds in Mathematical Physics (Week 124)</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week126.html">(Week 126)</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week147.html">(Week 147)</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week213.html">(Week 213)</a> <ul><li><a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf">Euler's Proof That 1 + 2 + 3 + ⋯ = −1/12</a> – by <a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Baez2008" class="citation web cs1"><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a> (September 19, 2008). <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/numbers/24.pdf">"My Favorite Numbers: 24"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=My+Favorite+Numbers%3A+24&rft.date=2008-09-19&rft.au=John+Baez&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fnumbers%2F24.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3A1+%2B+2+%2B+3+%2B+4+%2B+%E2%8B%AF" class="Z3988"></span></li></ul></li> <li><a rel="nofollow" class="external text" href="http://terrytao.wordpress.com/2010/04/10/the-euler-maclaurin-formula-bernoulli-numbers-the-zeta-function-and-real-variable-analytic-continuation/">The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation</a> by <a href="/wiki/Terence_Tao" title="Terence Tao">Terence Tao</a></li> <li><a rel="nofollow" class="external text" href="http://motls.blogspot.co.uk/2014/01/a-recursive-evaluation-of-zeta-of.html">A recursive evaluation of zeta of negative integers</a> by <a href="/wiki/Lubo%C5%A1_Motl" title="Luboš Motl">Luboš Motl</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=w-I6XTVZXww">Link to Numberphile video 1 + 2 + 3 + 4 + 5 + ... = –1/12</a> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=E-d9mgo8FGk">Sum of Natural Numbers (second proof and extra footage)</a> includes demonstration of Euler's method.</li> <li><a rel="nofollow" class="external text" href="http://www.nottingham.ac.uk/~ppzap4/response.html">What do we get if we sum all the natural numbers?</a> response to comments about video by Tony Padilla</li> <li><a rel="nofollow" class="external text" href="https://www.nytimes.com/2014/02/04/science/in-the-end-it-all-adds-up-to.html?hpw&rref=science">Related article from New York Times</a></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=0Oazb7IWzbA">Why –1/12 is a gold nugget</a> follow-up Numberphile video with <a href="/wiki/Edward_Frenkel" title="Edward Frenkel">Edward Frenkel</a></li></ul></li> <li><a rel="nofollow" class="external text" href="http://math.arizona.edu/~cais/Papers/Expos/div.pdf">Divergent Series: why 1 + 2 + 3 + ⋯ = −1/12</a> by Brydon Cais from University of Arizona</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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class="nv-edit"><a href="/wiki/Special:EditPage/Template:Series_(mathematics)" title="Special:EditPage/Template:Series (mathematics)"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Sequences_and_series" style="font-size:114%;margin:0 4em"><a href="/wiki/Sequence" title="Sequence">Sequences</a> and <a href="/wiki/Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li> <li><a href="/wiki/Geometric_progression" title="Geometric progression">Geometric progression</a></li> <li><a href="/wiki/Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li> <li><a href="/wiki/Square_number" title="Square number">Square number</a></li> <li><a href="/wiki/Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li> <li><a href="/wiki/Factorial" title="Factorial">Factorial</a></li> <li><a href="/wiki/Power_of_two" title="Power of two">Powers of two</a></li> <li><a href="/wiki/Power_of_three" title="Power of three">Powers of three</a></li> <li><a href="/wiki/Power_of_10" title="Power of 10">Powers of 10</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="/wiki/List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Complete_sequence" title="Complete sequence">Complete sequence</a></li> <li><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li> <li><a href="/wiki/Figurate_number" title="Figurate number">Figurate number</a></li> <li><a href="/wiki/Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li> <li><a href="/wiki/Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li> <li><a href="/wiki/Lucas_number" title="Lucas number">Lucas number</a></li> <li><a href="/wiki/Pell_number" title="Pell number">Pell number</a></li> <li><a href="/wiki/Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li> <li><a href="/wiki/Polygonal_number" title="Polygonal number">Polygonal number</a></li> <li><a href="/wiki/Triangular_number" title="Triangular number">Triangular number</a> <ul><li><a href="/wiki/Triangular_array" title="Triangular array">array</a></li></ul></li></ul> </div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/Fibonacci_sequence" title="Fibonacci sequence"><img alt="Fibonacci spiral with square sizes up to 34." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/80px-Fibonacci_spiral_34.svg.png" decoding="async" width="80" height="51" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/120px-Fibonacci_spiral_34.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Fibonacci_spiral_34.svg/160px-Fibonacci_spiral_34.svg.png 2x" data-file-width="915" data-file-height="579" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li> <li><a href="/wiki/Monotonic_function" title="Monotonic function">Monotonic function</a></li> <li><a href="/wiki/Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Convergent_series" title="Convergent series">Convergent</a></li> <li><a href="/wiki/Divergent_series" title="Divergent series">Divergent</a></li> <li><a href="/wiki/Telescoping_series" title="Telescoping series">Telescoping</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_convergence" title="Absolute convergence">Absolute</a></li> <li><a href="/wiki/Conditional_convergence" title="Conditional convergence">Conditional</a></li> <li><a href="/wiki/Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1/2_%E2%88%92_1/4_%2B_1/8_%E2%88%92_1/16_%2B_%E2%8B%AF" title="1/2 − 1/4 + 1/8 − 1/16 + ⋯">1/2 − 1/4 + 1/8 − 1/16 + ⋯</a></li> <li><a href="/wiki/1/2_%2B_1/4_%2B_1/8_%2B_1/16_%2B_%E2%8B%AF" title="1/2 + 1/4 + 1/8 + 1/16 + ⋯">1/2 + 1/4 + 1/8 + 1/16 + ⋯</a></li> <li><a href="/wiki/1/4_%2B_1/16_%2B_1/64_%2B_1/256_%2B_%E2%8B%AF" title="1/4 + 1/16 + 1/64 + 1/256 + ⋯">1/4 + 1/16 + 1/64 + 1/256 + ⋯</a></li> <li><a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/1_%2B_1_%2B_1_%2B_1_%2B_%E2%8B%AF" title="1 + 1 + 1 + 1 + ⋯">1 + 1 + 1 + 1 + ⋯</a></li> <li><a href="/wiki/Grandi%27s_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a class="mw-selflink selflink">1 + 2 + 3 + 4 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF" title="1 − 2 + 3 − 4 + ⋯">1 − 2 + 3 − 4 + ⋯</a></li> <li><a href="/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B%AF" title="1 + 2 + 4 + 8 + ⋯">1 + 2 + 4 + 8 + ⋯</a></li> <li><a href="/wiki/1_%E2%88%92_2_%2B_4_%E2%88%92_8_%2B_%E2%8B%AF" title="1 − 2 + 4 − 8 + ⋯">1 − 2 + 4 − 8 + ⋯</a></li> <li><a href="/wiki/Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li> <li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_..." class="mw-redirect" title="1 − 1 + 2 − 6 + 24 − 120 + ...">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li> <li><a href="/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a></li> <li><a href="/wiki/Power_series" title="Power series">Power series</a></li> <li><a href="/wiki/Formal_power_series" title="Formal power series">Formal power series</a></li> <li><a href="/wiki/Laurent_series" title="Laurent series">Laurent series</a></li> <li><a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a></li> <li><a href="/wiki/Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li> <li><a href="/wiki/Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li> <li><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></li> <li><a href="/wiki/Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li> <li><a href="/wiki/Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li> <li><a href="/wiki/Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li> <li><a href="/wiki/Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li> <li><a href="/wiki/Riemann%27s_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li> <li><a href="/wiki/Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Mathematical_series" title="Category:Mathematical series">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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