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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Education"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Education</span> </div> </a> <button aria-controls="toc-Education-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 Education subsection</span> </button> <ul id="toc-Education-sublist" class="vector-toc-list"> <li id="toc-Cognitive_impact" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cognitive_impact"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Cognitive impact</span> </div> </a> <ul id="toc-Cognitive_impact-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Preconceptions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Preconceptions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Preconceptions</span> </div> </a> <ul id="toc-Preconceptions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prospects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prospects"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Prospects</span> </div> </a> <ul id="toc-Prospects-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Summability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Summability"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Summability</span> </div> </a> <ul id="toc-Summability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_problems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_problems"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Related problems</span> </div> </a> <ul id="toc-Related_problems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">10</span> <span>References</span> </div> </a> <ul id="toc-References-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">11</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">Grandi's series</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B3%D9%84%D8%B3%D9%84%D8%A9_%D8%BA%D8%B1%D8%A7%D9%86%D8%AF%D9%8A" title="متسلسلة غراندي – Arabic" lang="ar" hreflang="ar" data-title="متسلسلة غراندي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qrandi_silsil%C9%99si" title="Qrandi silsiləsi – Azerbaijani" lang="az" hreflang="az" data-title="Qrandi silsiləsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A6%BE%E0%A6%A8%E0%A7%8D%E0%A6%A6%E0%A7%87%E0%A6%B0_%E0%A6%A7%E0%A6%BE%E0%A6%B0%E0%A6%BE" title="গ্রান্দের ধারা – Bangla" lang="bn" hreflang="bn" data-title="গ্রান্দের ধারা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/S%C3%A8rie_de_Grandi" title="Sèrie de Grandi – Catalan" lang="ca" hreflang="ca" data-title="Sèrie de Grandi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Grandi-Reihe" title="Grandi-Reihe – German" lang="de" hreflang="de" data-title="Grandi-Reihe" 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/Serie_de_Grandi" title="Serie de Grandi – Spanish" lang="es" hreflang="es" data-title="Serie de Grandi" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/S%C3%A9rie_de_Grandi" title="Série de Grandi – French" lang="fr" hreflang="fr" data-title="Série de Grandi" 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%A4%97%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%82%E0%A4%A1%E0%A5%80_%E0%A4%B6%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%A3%E0%A5%80" title="ग्रांडी श्रेणी – Hindi" lang="hi" hreflang="hi" data-title="ग्रांडी श्रेणी" 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/Deret_Grandi" title="Deret Grandi – Indonesian" lang="id" hreflang="id" data-title="Deret Grandi" 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/Serie_di_Grandi" title="Serie di Grandi – Italian" lang="it" hreflang="it" data-title="Serie di Grandi" 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%92%D7%A8%D7%A0%D7%93%D7%99" title="טור גרנדי – Hebrew" lang="he" hreflang="he" data-title="טור גרנדי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Grandi-reeks" title="Grandi-reeks – Dutch" lang="nl" hreflang="nl" data-title="Grandi-reeks" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B0%E3%83%A9%E3%83%B3%E3%83%87%E3%82%A3%E7%B4%9A%E6%95%B0" title="グランディ級数 – Japanese" lang="ja" hreflang="ja" data-title="グランディ級数" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Grandis_rekke" title="Grandis rekke – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Grandis rekke" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szereg_Grandiego" title="Szereg Grandiego – Polish" lang="pl" hreflang="pl" data-title="Szereg Grandiego" 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/S%C3%A9rie_de_Grandi" title="Série de Grandi – Portuguese" lang="pt" hreflang="pt" data-title="Série de Grandi" 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/Seria_lui_Grandi" title="Seria lui Grandi – Romanian" lang="ro" hreflang="ro" data-title="Seria lui Grandi" 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%93%D1%80%D0%B0%D0%BD%D0%B4%D0%B8" title="Ряд Гранди – Russian" lang="ru" hreflang="ru" data-title="Ряд Гранди" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://sl.wikipedia.org/wiki/Grandijeva_vrsta" title="Grandijeva vrsta – Slovenian" lang="sl" hreflang="sl" data-title="Grandijeva vrsta" 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/%D0%93%D1%80%D0%B0%D0%BD%D0%B4%D0%B8%D1%98%D0%B5%D0%B2%D0%B8_%D1%80%D0%B5%D0%B4%D0%BE%D0%B2%D0%B8" title="Грандијеви редови – Serbian" lang="sr" hreflang="sr" data-title="Грандијеви редови" 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Grandin_sarja" title="Grandin sarja – Finnish" lang="fi" hreflang="fi" data-title="Grandin sarja" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Grandis_serie" title="Grandis serie – Swedish" lang="sv" hreflang="sv" data-title="Grandis serie" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AD%E0%B8%99%E0%B8%B8%E0%B8%81%E0%B8%A3%E0%B8%A1%E0%B8%81%E0%B8%A3%E0%B8%B1%E0%B8%99%E0%B8%94%E0%B8%B5" title="อนุกรมกรันดี – Thai" lang="th" hreflang="th" data-title="อนุกรมกรันดี" 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/Grandi_serisi" title="Grandi serisi – Turkish" lang="tr" hreflang="tr" data-title="Grandi serisi" 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/%D0%A0%D1%8F%D0%B4_%D0%93%D1%80%D0%B0%D0%BD%D0%B4%D1%96" title="Ряд Гранді – Ukrainian" lang="uk" hreflang="uk" data-title="Ряд Гранді" 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/Chu%E1%BB%97i_Grandi" title="Chuỗi Grandi – Vietnamese" lang="vi" hreflang="vi" data-title="Chuỗi Grandi" 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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A0%BC%E8%98%AD%E8%BF%AA%E7%B4%9A%E6%95%B8" title="格蘭迪級數 – Chinese" lang="zh" hreflang="zh" data-title="格蘭迪級數" 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 wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q967588#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">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">Infinite series summing alternating 1 and -1 terms</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</a> <span class="nowrap">1 − 1 + 1 − 1 + ⋯</span>, also written </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 }(-1)^{n}}"> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ed1c55a1ac34c32acb7e1c72b2176f8b791448a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.353ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}}"></span></dd></dl> <p>is sometimes called <b>Grandi's series</b>, after Italian <a href="/wiki/Mathematician" title="Mathematician">mathematician</a>, <a href="/wiki/Philosopher" class="mw-redirect" title="Philosopher">philosopher</a>, and priest <a href="/wiki/Guido_Grandi" class="mw-redirect" title="Guido Grandi">Guido Grandi</a>, who gave a memorable treatment of the series in 1703. It is a <a href="/wiki/Divergent_series" title="Divergent series">divergent series</a>, meaning that the sequence of partial sums of the series does not converge. </p><p>However, though it is divergent, it can be manipulated to yield a number of mathematically interesting 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. For example, the <a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a> and the <a href="/wiki/Ramanujan_summation" title="Ramanujan summation">Ramanujan summation</a> of this series are both 1/2. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Nonrigorous_methods">Nonrigorous methods</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=1" title="Edit section: Nonrigorous methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One obvious method to find the sum of the series </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+1-1+1-1+1-1+\ldots }"> <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> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-1+1-1+1-1+1-1+\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6d6317cf1d4c3095c6a528c4c31c47ed89ed1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:34.746ex; height:2.343ex;" alt="{\displaystyle 1-1+1-1+1-1+1-1+\ldots }"></span></dd></dl> <p>would be to treat it like a <a href="/wiki/Telescoping_series" title="Telescoping series">telescoping series</a> and perform the subtractions in place: </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)+(1-1)+(1-1)+(1-1)+\ldots =0+0+0+0+\ldots =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>=</mo> <mn>0</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-1)+(1-1)+(1-1)+(1-1)+\ldots =0+0+0+0+\ldots =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5cfea515666b655c24b4415d5126f14d761b23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.724ex; height:2.843ex;" alt="{\displaystyle (1-1)+(1-1)+(1-1)+(1-1)+\ldots =0+0+0+0+\ldots =0.}"></span></dd></dl> <p>On the other hand, a similar bracketing procedure leads to the apparently contradictory result </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)+(-1+1)+(-1+1)+\ldots =1+0+0+0+\ldots =1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> <mo>=</mo> <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>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+\ldots =1+0+0+0+\ldots =1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4182f54f47077eac0a75b13ae6e14aecf7c3e576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.336ex; height:2.843ex;" alt="{\displaystyle 1+(-1+1)+(-1+1)+(-1+1)+\ldots =1+0+0+0+\ldots =1.}"></span></dd></dl> <p>Thus, by applying parentheses to Grandi's series in different ways, one can obtain either 0 or 1 as a "value". This is closely akin to the general problem of <a href="/wiki/Conditional_convergence" title="Conditional convergence">conditional convergence</a>, and variations of this idea, called the <a href="/wiki/Eilenberg%E2%80%93Mazur_swindle" title="Eilenberg–Mazur swindle">Eilenberg–Mazur swindle</a>, are sometimes used in <a href="/wiki/Knot_theory" title="Knot theory">knot theory</a> and <a href="/wiki/Algebra" title="Algebra">algebra</a>. By taking the average of these two "values", one can justify that the series converges to 1/2. </p><p>Treating Grandi's series as a <a href="/wiki/Divergent_geometric_series" title="Divergent geometric series">divergent geometric series</a> and using the same algebraic methods that evaluate convergent geometric series to obtain a third value: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S&=1-1+1-1+\ldots ,{\text{ so}}\\1-S&=1-(1-1+1-1+\ldots )=1-1+1-1+\ldots =S\\1-S&=S\\1&=2S,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> so</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mo>…</mo> <mo>=</mo> <mi>S</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>−</mo> <mi>S</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>S</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S&=1-1+1-1+\ldots ,{\text{ so}}\\1-S&=1-(1-1+1-1+\ldots )=1-1+1-1+\ldots =S\\1-S&=S\\1&=2S,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d432efd0c85c68a8f157ff41bf7bbf0393d85bf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:60.33ex; height:12.009ex;" alt="{\displaystyle {\begin{aligned}S&=1-1+1-1+\ldots ,{\text{ so}}\\1-S&=1-(1-1+1-1+\ldots )=1-1+1-1+\ldots =S\\1-S&=S\\1&=2S,\end{aligned}}}"></span> </p><p>resulting in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f177c399f958129671c4a243ef4844b8bcfad0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.085ex; height:2.843ex;" alt="{\displaystyle S=1/2}"></span>. The same conclusion results from calculating <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 -S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f1e4d54025d9e28e36c3cced3cc0dd7c04da43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.307ex; height:2.343ex;" alt="{\textstyle -S}"></span> (from (<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 -S=(1-S)-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>−</mo> <mi>S</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle -S=(1-S)-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4094558f02ea11b03b286b88d9162d17bcf8ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.72ex; height:2.843ex;" alt="{\textstyle -S=(1-S)-1}"></span>), subtracting the result from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, and solving <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 2S=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2S=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42af43456781336c4d01c2323c2e56dbd4051f71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.923ex; height:2.176ex;" alt="{\displaystyle 2S=1}"></span>.<sup id="cite_ref-FOOTNOTEDevlin199477_1-0" class="reference"><a href="#cite_note-FOOTNOTEDevlin199477-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The above manipulations do not consider what the sum of a series rigorously means and how said algebraic methods can be applied to <a href="/wiki/Divergent_geometric_series" title="Divergent geometric series">divergent geometric series</a>. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions: </p> <ul><li>The series <span class="nowrap">1 − 1 + 1 − 1 + ...</span> has no sum.<sup id="cite_ref-FOOTNOTEDevlin199477_1-1" class="reference"><a href="#cite_note-FOOTNOTEDevlin199477-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEDavis1989152_2-0" class="reference"><a href="#cite_note-FOOTNOTEDavis1989152-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>... but its sum <i>should</i> be 1/2.<sup id="cite_ref-FOOTNOTEDavis1989152_2-1" class="reference"><a href="#cite_note-FOOTNOTEDavis1989152-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul> <p>In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century <a href="/wiki/History_of_calculus" title="History of calculus">introduction of calculus in Europe</a>, but before the advent of modern <a href="/wiki/Rigor#Mathematical_rigour" class="mw-redirect" title="Rigor">rigour</a>, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between <a href="/wiki/Mathematician" title="Mathematician">mathematicians</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> <div class="mw-heading mw-heading2"><h2 id="Relation_to_the_geometric_series">Relation to the geometric series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=2" title="Edit section: Relation to the geometric series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> in the interval <span class="nowrap">⁠<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)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e120a3bd60fc89b495dd7ef6039465b7e6a703b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,1)}"></span>⁠</span>, the <a href="/wiki/Geometric_progression#Series" title="Geometric progression">sum to infinity of a geometric series</a> can be evaluated via </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1420b6ce0842d437538e123e5a2eabca918e4b22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.882ex; height:7.343ex;" alt="{\displaystyle \lim _{N\to \infty }\sum _{n=0}^{N}r^{n}=\sum _{n=0}^{\infty }r^{n}={\frac {1}{1-r}}.}"></span></dd></dl> <p>For any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \in (0,2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon \in (0,2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/539cdd4a7ac0ed0e65fca63c44cb383d34e06600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.092ex; height:2.843ex;" alt="{\displaystyle \varepsilon \in (0,2)}"></span>, one thus finds </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 }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}"> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>−</mo> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c00904b22e59f085acd99def864598ba87c4a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.586ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(-1+\varepsilon )^{n}={\frac {1}{1-(-1+\varepsilon )}}={\frac {1}{2-\varepsilon }},}"></span></dd></dl> <p>and so the limit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon \to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon \to 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a6823c23666f99317e232cf7d02df6d9c9b7a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.86ex; height:2.176ex;" alt="{\displaystyle \varepsilon \to 0}"></span> of series evaluations is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652c8c9e95fcac5cedf1a88026af0be8dcf5fe08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.772ex; height:7.343ex;" alt="{\displaystyle \lim _{\varepsilon \to 0}\lim _{N\to \infty }\sum _{n=0}^{N}(-1+\varepsilon )^{n}={\frac {1}{2}}.}"></span></dd></dl> <p>However, as mentioned, the series obtained by switching the limits, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ε</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <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> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>ε</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e330ae5edd96cd453114aa5094a1f29a410abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.48ex; height:7.343ex;" alt="{\displaystyle \lim _{N\to \infty }\lim _{\varepsilon \to 0}\sum _{n=0}^{N}(-1+\varepsilon )^{n}=\sum _{n=0}^{\infty }(-1)^{n}}"></span></dd></dl> <p>is divergent. </p><p>In the terms of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <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="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span> is thus seen to be the value at <span class="nowrap"><i>z</i> = −1</span> of the <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> of the series <span class="nowrap">⁠<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}z^{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> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{n=0}^{N}z^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e878d40248c2bf7283dd22493e85008c3ed0faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.469ex; height:3.509ex;" alt="{\displaystyle \textstyle \sum _{n=0}^{N}z^{n}}"></span>⁠</span>, which is only defined on the complex unit disk, <span class="nowrap">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>z</i></span>| < 1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Early_ideas">Early ideas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=3" title="Edit section: Early ideas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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/History_of_Grandi%27s_series" title="History of Grandi's series">History of Grandi's series</a></div> <div class="mw-heading mw-heading2"><h2 id="Divergence">Divergence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=4" title="Edit section: Divergence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In modern mathematics, the sum of an infinite series is defined to be the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit of the sequence</a> of its <a href="/wiki/Partial_sum" class="mw-redirect" title="Partial sum">partial sums</a>, if it exists. The sequence of partial sums of Grandi's series is <span class="nowrap">1, 0, 1, 0, ...,</span> which clearly does not approach any number (although it does have two <a href="/wiki/Accumulation_point" title="Accumulation point">accumulation points</a> at 0 and 1). Therefore, Grandi's series is <a href="/wiki/Divergent_series" title="Divergent series">divergent</a>. </p><p>It can be shown that it is not valid to perform many seemingly innocuous operations on a series, such as reordering individual terms, unless the series is <a href="/wiki/Absolute_convergence" title="Absolute convergence">absolutely convergent</a>. Otherwise these operations can alter the result of summation.<sup id="cite_ref-FOOTNOTEProtterMorrey1991_4-0" class="reference"><a href="#cite_note-FOOTNOTEProtterMorrey1991-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Further, the terms of Grandi's series can be rearranged to have its accumulation points at any interval of two or more consecutive integer numbers, not only 0 or 1. For instance, the series </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+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }"> <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> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d1be0ab6612178c565d4e4177f710469e1dfc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:70.772ex; height:2.343ex;" alt="{\displaystyle 1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1-\cdots }"></span></dd></dl> <p>(in which, after five initial +1 terms, the terms alternate in pairs of +1 and −1 terms – the infinitude of both +1s and −1s allows any finite number of 1s or −1s to be prepended, by <a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's paradox of the Grand Hotel</a>) is a <a href="/wiki/Permutation" title="Permutation">permutation</a> of Grandi's series in which each value in the rearranged series corresponds to a value that is at most four positions away from it in the original series; its accumulation points are 3, 4, and 5. </p> <div class="mw-heading mw-heading2"><h2 id="Education">Education</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=5" title="Edit section: Education"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Cognitive_impact">Cognitive impact</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=6" title="Edit section: Cognitive impact"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Around 1987, <a href="/wiki/Anna_Sierpi%C5%84ska" title="Anna Sierpińska">Anna Sierpińska</a> introduced Grandi's series to a group of 17-year-old precalculus students at a <a href="/wiki/Warsaw" title="Warsaw">Warsaw</a> <a href="/wiki/Lyceum" title="Lyceum">lyceum</a>. She focused on humanities students with the expectation that their mathematical experience would be less significant than that of their peers studying mathematics and physics, so the <a href="/wiki/Epistemology" title="Epistemology">epistemological</a> obstacles they exhibit would be more representative of the obstacles that <i>may</i> still be present in lyceum students. </p><p>Sierpińska initially expected the students to balk at assigning a value to Grandi's series, at which point she could shock them by claiming that <span class="nowrap">1 − 1 + 1 − 1 + ··· = <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></span> as a result of the geometric series formula. Ideally, by searching for the error in reasoning and by investigating the formula for various common ratios, the students would "notice that there are two kinds of series and an implicit conception of convergence will be born".<sup id="cite_ref-FOOTNOTESierpińska1987371–378_5-0" class="reference"><a href="#cite_note-FOOTNOTESierpińska1987371–378-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> However, the students showed no shock at being told that <span class="nowrap">1 − 1 + 1 − 1 + ··· = <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></span> or even that <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> = −1</span>. Sierpińska remarks that <i>a priori</i>, the students' reaction shouldn't be too surprising given that Leibniz and Grandi thought <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> to be a plausible result; </p> <dl><dd>"A posteriori, however, the explanation of this lack of shock on the part of the students may be somewhat different. They accepted calmly the absurdity because, after all, 'mathematics is completely abstract and far from reality', and 'with those mathematical transformations you can prove all kinds of nonsense', as one of the boys later said."<sup id="cite_ref-FOOTNOTESierpińska1987371–378_5-1" class="reference"><a href="#cite_note-FOOTNOTESierpińska1987371–378-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The students were ultimately not immune to the question of convergence; Sierpińska succeeded in engaging them in the issue by linking it to decimal expansions the following day. As soon as <span class="nowrap"><a href="/wiki/0.999..." title="0.999...">0.999...</a> = 1</span> caught the students by surprise, the rest of her material "went past their ears".<sup id="cite_ref-FOOTNOTESierpińska1987371–378_5-2" class="reference"><a href="#cite_note-FOOTNOTESierpińska1987371–378-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Preconceptions">Preconceptions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=7" title="Edit section: Preconceptions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In another study conducted in <a href="/wiki/Treviso" title="Treviso">Treviso</a>, <a href="/wiki/Italy" title="Italy">Italy</a> around the year 2000, third-year and fourth-year <i><a href="/wiki/Liceo_Scientifico" class="mw-redirect" title="Liceo Scientifico">Liceo Scientifico</a></i> pupils (between 16 and 18 years old) were given cards asking the following: </p> <dl><dd>"In 1703, the mathematician Guido Grandi studied the addition: <span class="nowrap">1 − 1 + 1 − 1 + ...</span> (addends, infinitely many, are always +1 and –1). What is your opinion about it?"</dd></dl> <p>The students had been introduced to the idea of an infinite set, but they had no prior experience with infinite series. They were given ten minutes without books or calculators. The 88 responses were categorized as follows: </p> <dl><dd>(26) the result is 0</dd> <dd>(18) the result can be either 0 or 1</dd> <dd>(5) the result does not exist</dd> <dd>(4) the result is <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></dd> <dd>(3) the result is 1</dd> <dd>(2) the result is infinite</dd> <dd>(30) no answer</dd></dl> <p>The researcher, Giorgio Bagni, interviewed several of the students to determine their reasoning. Some 16 of them justified an answer of 0 using logic similar to that of Grandi and Riccati. Others justified <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> as being the average of 0 and 1. Bagni notes that their reasoning, while similar to Leibniz's, lacks the probabilistic basis that was so important to 18th-century mathematics. He concludes that the responses are consistent with a link between historical development and individual development, although the cultural context is different.<sup id="cite_ref-FOOTNOTEBagni20056–8_6-0" class="reference"><a href="#cite_note-FOOTNOTEBagni20056–8-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Prospects">Prospects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=8" title="Edit section: Prospects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Joel Lehmann describes the process of distinguishing between different sum concepts as building a bridge over a conceptual crevasse: the confusion over divergence that dogged 18th-century mathematics. </p> <dl><dd>"Since series are generally presented without history and separate from applications, the student must wonder not only "What are these things?" but also "Why are we doing this?" The preoccupation with determining convergence but not the sum makes the whole process seem artificial and pointless to many students—and instructors as well."<sup id="cite_ref-FOOTNOTELehmann1995165_7-0" class="reference"><a href="#cite_note-FOOTNOTELehmann1995165-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></dd></dl> <p>As a result, many students develop an attitude similar to Euler's: </p> <dl><dd>"... problems that arise naturally (i.e., from nature) do have solutions, so the assumption that things will work out eventually is justified experimentally without the need for existence sorts of proof. Assume everything is okay, and if the arrived-at solution works, you were probably right, or at least right enough. ... so why bother with the details that only show up in homework problems?"<sup id="cite_ref-FOOTNOTELehmann1995176_8-0" class="reference"><a href="#cite_note-FOOTNOTELehmann1995176-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></dd></dl> <p>Lehmann recommends meeting this objection with the same example that was advanced against Euler's treatment of Grandi's series by Jean-Charles Callet. Euler had viewed the sum as the evaluation at <span class="nowrap"><i>x</i> = 1</span> of the <a href="/wiki/Geometric_series" title="Geometric series">geometric series</a> <span class="nowrap">⁠<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-x+x^{2}-x^{3}+\cdots =1/(1+x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-x+x^{2}-x^{3}+\cdots =1/(1+x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/861ff6fcf01aefc7b1f8f7dbb153ce85ee9c4520" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.91ex; height:3.176ex;" alt="{\displaystyle 1-x+x^{2}-x^{3}+\cdots =1/(1+x)}"></span>⁠</span>, giving the sum <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>. However, Callet pointed out that one could instead view Grandi's series as the evaluation at <span class="nowrap"><i>x</i> = 1</span> of a different series, <span class="nowrap">⁠<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-x^{2}+x^{3}-x^{5}+x^{6}-\cdots ={\tfrac {1+x}{1+x+x^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-x^{2}+x^{3}-x^{5}+x^{6}-\cdots ={\tfrac {1+x}{1+x+x^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686fddc0f9ea12246412c10fd194d5e87a4c397b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.649ex; height:4.176ex;" alt="{\displaystyle 1-x^{2}+x^{3}-x^{5}+x^{6}-\cdots ={\tfrac {1+x}{1+x+x^{2}}}}"></span>⁠</span>, giving the sum <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>. Lehman argues that seeing such a conflicting outcome in intuitive evaluations may motivate the need for rigorous definitions and attention to detail.<sup id="cite_ref-FOOTNOTELehmann1995176_8-1" class="reference"><a href="#cite_note-FOOTNOTELehmann1995176-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </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=Grandi%27s_series&action=edit&section=9" title="Edit section: Summability"><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">Main article: <a href="/wiki/Summation_of_Grandi%27s_series" title="Summation of Grandi's series">Summation of Grandi's series</a></div> <div class="mw-heading mw-heading2"><h2 id="Related_problems">Related problems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=10" title="Edit section: Related problems"><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">Main article: <a href="/wiki/Occurrences_of_Grandi%27s_series" title="Occurrences of Grandi's series">Occurrences of Grandi's series</a></div> <p>The series <span class="nowrap">1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ...</span> (<a href="/wiki/Up_to" title="Up to">up to</a> infinity) is also divergent, but some methods may be used to sum it 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>. This is the square of the value most summation methods assign to Grandi's series, which is reasonable as it can be viewed as the <a href="/wiki/Cauchy_product" title="Cauchy product">Cauchy product</a> of two copies of Grandi's series. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯</a></li> <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/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_3_%2B_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/Ramanujan_summation" title="Ramanujan summation">Ramanujan summation</a></li> <li><a href="/wiki/Ces%C3%A0ro_summation" title="Cesàro summation">Cesàro summation</a></li> <li><a href="/wiki/Thomson%27s_lamp" title="Thomson's lamp">Thomson's lamp</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grandi%27s_series&action=edit&section=12" title="Edit section: Notes"><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 reflist-columns-2"> <ol class="references"> <li id="cite_note-FOOTNOTEDevlin199477-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDevlin199477_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDevlin199477_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDevlin1994">Devlin 1994</a>, p. 77</span> </li> <li id="cite_note-FOOTNOTEDavis1989152-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDavis1989152_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDavis1989152_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDavis1989">Davis 1989</a>, p. 152</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFKline1983">Kline 1983</a>, p. 307; <a href="#CITEREFKnopp1990">Knopp 1990</a>, p. 457</span> </li> <li id="cite_note-FOOTNOTEProtterMorrey1991-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEProtterMorrey1991_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFProtterMorrey1991">Protter & Morrey 1991</a></span> </li> <li id="cite_note-FOOTNOTESierpińska1987371–378-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESierpińska1987371–378_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESierpińska1987371–378_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTESierpińska1987371–378_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSierpińska1987">Sierpińska 1987</a>, pp. 371–378</span> </li> <li id="cite_note-FOOTNOTEBagni20056–8-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBagni20056–8_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBagni2005">Bagni 2005</a>, pp. 6–8</span> </li> <li id="cite_note-FOOTNOTELehmann1995165-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELehmann1995165_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLehmann1995">Lehmann 1995</a>, p. 165</span> </li> <li id="cite_note-FOOTNOTELehmann1995176-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTELehmann1995176_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTELehmann1995176_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFLehmann1995">Lehmann 1995</a>, p. 176</span> </li> </ol></div> <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=Grandi%27s_series&action=edit&section=13" title="Edit section: References"><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><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription 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 .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBagni2005" class="citation journal cs1">Bagni, Giorgio T. (June 2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20061229024056/http://www.cimt.plymouth.ac.uk/journal/bagni.pdf">"Infinite Series from History to Mathematics Education"</a> <span class="cs1-format">(PDF)</span>. <i>International Journal for Mathematics Teaching and Learning</i>. Archived from <a rel="nofollow" class="external text" href="http://www.cimt.plymouth.ac.uk/journal/bagni.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-12-29.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+for+Mathematics+Teaching+and+Learning&rft.atitle=Infinite+Series+from+History+to+Mathematics+Education&rft.date=2005-06&rft.aulast=Bagni&rft.aufirst=Giorgio+T.&rft_id=http%3A%2F%2Fwww.cimt.plymouth.ac.uk%2Fjournal%2Fbagni.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavis1989" class="citation book cs1">Davis, Harry F. (May 1989). <i>Fourier Series and Orthogonal Functions</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65973-2" title="Special:BookSources/978-0-486-65973-2"><bdi>978-0-486-65973-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Series+and+Orthogonal+Functions&rft.pub=Dover&rft.date=1989-05&rft.isbn=978-0-486-65973-2&rft.aulast=Davis&rft.aufirst=Harry+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDevlin1994" class="citation book cs1"><a href="/wiki/Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicsscien0000devl"><i>Mathematics, the science of patterns: the search for order in life, mind, and the universe</i></a></span>. Scientific American Library. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-6022-1" title="Special:BookSources/978-0-7167-6022-1"><bdi>978-0-7167-6022-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics%2C+the+science+of+patterns%3A+the+search+for+order+in+life%2C+mind%2C+and+the+universe&rft.pub=Scientific+American+Library&rft.date=1994&rft.isbn=978-0-7167-6022-1&rft.aulast=Devlin&rft.aufirst=Keith&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicsscien0000devl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1983" class="citation journal cs1">Kline, Morris (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 title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Euler+and+Infinite+Series&rft.volume=56&rft.issue=5&rft.pages=307-314&rft.date=1983-11&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.639.6923%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690371%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2690371&rft.aulast=Kline&rft.aufirst=Morris&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><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]. <i>Theory and Application of Infinite Series</i>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-66165-0" title="Special:BookSources/978-0-486-66165-0"><bdi>978-0-486-66165-0</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.pub=Dover&rft.date=1990&rft.isbn=978-0-486-66165-0&rft.aulast=Knopp&rft.aufirst=Konrad&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHobson1907" class="citation book cs1">Hobson, E. W. (1907). <a rel="nofollow" class="external text" href="https://archive.org/details/theoryfunctionsr00hobs"><i>The theory of functions of a real variable and the theory of Fourier's series</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. section 331. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4181-8651-7" title="Special:BookSources/978-1-4181-8651-7"><bdi>978-1-4181-8651-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+theory+of+functions+of+a+real+variable+and+the+theory+of+Fourier%27s+series&rft.pages=section+331&rft.pub=Cambridge+University+Press&rft.date=1907&rft.isbn=978-1-4181-8651-7&rft.aulast=Hobson&rft.aufirst=E.+W.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftheoryfunctionsr00hobs&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLehmann1995" class="citation conference cs1">Lehmann, Joel (1995). "Converging Concepts of Series: Learning from History". In Swetz, Frank; Fauvel, John; Bekken, Otto; Johansson, Bengt; Katz, Victor (eds.). <a rel="nofollow" class="external text" href="https://faculty.uml.edu/cbyrne/Lehmann.pdf"><i>Learn from the Masters!</i></a> <span class="cs1-format">(PDF)</span>. Mathematical Association of America. pp. 161–180.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=Converging+Concepts+of+Series%3A+Learning+from+History&rft.btitle=Learn+from+the+Masters%21&rft.pages=161-180&rft.pub=Mathematical+Association+of+America&rft.date=1995&rft.aulast=Lehmann&rft.aufirst=Joel&rft_id=https%3A%2F%2Ffaculty.uml.edu%2Fcbyrne%2FLehmann.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProtterMorrey1991" class="citation book cs1">Protter, Murray H.; Morrey, Charles B. Jr. (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0pZJeQ2lEmkC&pg=PA249"><i>A First Course in Real Analysis</i></a>. <a href="/wiki/Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. Springer. p. 249. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97437-8" title="Special:BookSources/978-0-387-97437-8"><bdi>978-0-387-97437-8</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+Real+Analysis&rft.series=Undergraduate+Texts+in+Mathematics&rft.pages=249&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-97437-8&rft.aulast=Protter&rft.aufirst=Murray+H.&rft.au=Morrey%2C+Charles+B.+Jr.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0pZJeQ2lEmkC%26pg%3DPA249&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSierpińska1987" class="citation journal cs1"><a href="/wiki/Anna_Sierpi%C5%84ska" title="Anna Sierpińska">Sierpińska, Anna</a> (November 1987). "Humanities students and epistemological obstacles related to limits". <i>Educational Studies in Mathematics</i>. <b>18</b> (4): 371–396. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00240986">10.1007/BF00240986</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/3482354">3482354</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:144880659">144880659</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Educational+Studies+in+Mathematics&rft.atitle=Humanities+students+and+epistemological+obstacles+related+to+limits&rft.volume=18&rft.issue=4&rft.pages=371-396&rft.date=1987-11&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A144880659%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3482354%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1007%2FBF00240986&rft.aulast=Sierpi%C5%84ska&rft.aufirst=Anna&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittakerWatson1962" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Whittaker, E. T.</a>; <a href="/wiki/G._N._Watson" title="G. N. Watson">Watson, G. N.</a> (1962). <i><a href="/wiki/A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a></i> (4th, reprinted ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. § 2.1.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+of+Modern+Analysis&rft.pages=%C2%A7+2.1&rft.edition=4th%2C+reprinted&rft.pub=Cambridge+University+Press&rft.date=1962&rft.aulast=Whittaker&rft.aufirst=E.+T.&rft.au=Watson%2C+G.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGrandi%27s+series" 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=Grandi%27s_series&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=PCu_BNNI5x4">One minus one plus one minus one – Numberphile</a>, Grandi's series</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 ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Sequences_and_series" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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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 class="mw-selflink selflink">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li> <li><a href="/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF" title="1 + 2 + 3 + 4 + ⋯">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 class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Grandi&#039;s_series" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Grandi%27s_series" title="Template:Grandi's series"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Grandi%27s_series" title="Template talk:Grandi's series"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Grandi%27s_series" title="Special:EditPage/Template:Grandi's series"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Grandi&#039;s_series" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Grandi's series</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_Grandi%27s_series" title="History of Grandi's series">History</a></li> <li><a href="/wiki/Grandi%27s_series_in_education" class="mw-redirect" title="Grandi's series in education">Education</a></li> <li><a href="/wiki/Summation_of_Grandi%27s_series" title="Summation of Grandi's series">Summation</a></li> <li><a href="/wiki/Occurrences_of_Grandi%27s_series" title="Occurrences of Grandi's series">Occurrences</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</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/Luigi_Guido_Grandi" title="Luigi Guido Grandi">Luigi Guido Grandi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Thomson%27s_lamp" title="Thomson's lamp">Thomson's lamp</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><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:Grandi%27s_series" title="Category:Grandi's series">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img 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