디리클레 판정법 - 위키백과, 우리 모두의 백과사전

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id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-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>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-실수_항_급수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#실수_항_급수"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>실수 항 급수</span> </div> </a> <ul id="toc-실수_항_급수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-이상_적분" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#이상_적분"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>이상 적분</span> </div> </a> <ul id="toc-이상_적분-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-균등_수렴" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#균등_수렴"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>균등 수렴</span> </div> </a> <ul id="toc-균등_수렴-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>예</span> </div> </a> <button aria-controls="toc-예-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>예 하위섹션 토글하기</span> </button> <ul id="toc-예-sublist" class="vector-toc-list"> <li id="toc-교대급수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#교대급수"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>교대급수</span> </div> </a> <ul id="toc-교대급수-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-삼각_급수" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#삼각_급수"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>삼각 급수</span> </div> </a> <ul id="toc-삼각_급수-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-역사" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>역사</span> </div> </a> <ul id="toc-역사-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-각주" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#각주"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>각주</span> </div> </a> <ul id="toc-각주-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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="목차" 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="목차 토글" > <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">목차 토글</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">디리클레 판정법</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="다른 언어로 문서를 방문합니다. 18개 언어로 읽을 수 있습니다" > <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-18" 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">18개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D8%AE%D8%AA%D8%A8%D8%A7%D8%B1_%D8%AF%D9%8A%D8%B1%D9%8A%D9%83%D9%84%D9%8A%D9%87" title="اختبار ديريكليه – 아랍어" lang="ar" hreflang="ar" data-title="اختبار ديريكليه" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Dirichletov_test" title="Dirichletov test – 보스니아어" lang="bs" hreflang="bs" data-title="Dirichletov test" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kriterium_von_Dirichlet" title="Kriterium von Dirichlet – 독일어" lang="de" hreflang="de" data-title="Kriterium von Dirichlet" data-language-autonym="Deutsch" data-language-local-name="독일어" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Dirichlet%27s_test" title="Dirichlet's test – 영어" lang="en" hreflang="en" data-title="Dirichlet's test" data-language-autonym="English" data-language-local-name="영어" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Test_de_Dirichlet" title="Test de Dirichlet – 프랑스어" lang="fr" hreflang="fr" data-title="Test de Dirichlet" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%91%D7%97%D7%9F_%D7%93%D7%99%D7%A8%D7%99%D7%9B%D7%9C%D7%94" title="מבחן דיריכלה – 히브리어" lang="he" hreflang="he" data-title="מבחן דיריכלה" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%A1%E0%A5%80%E0%A4%B0%E0%A4%BF%E0%A4%96%E0%A5%8D%E0%A4%B2%E0%A5%87_%E0%A4%AA%E0%A4%B0%E0%A5%80%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A3" title="डीरिख्ले परीक्षण – 힌디어" lang="hi" hreflang="hi" data-title="डीरिख्ले परीक्षण" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Criterio_di_Dirichlet_(matematica)" title="Criterio di Dirichlet (matematica) – 이탈리아어" lang="it" hreflang="it" data-title="Criterio di Dirichlet (matematica)" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%87%E3%82%A3%E3%83%AA%E3%82%AF%E3%83%AC%E3%81%AE%E5%88%A4%E5%AE%9A%E6%B3%95" title="ディリクレの判定法 – 일본어" lang="ja" hreflang="ja" data-title="ディリクレの判定法" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%80%D0%B8%D1%85%D0%BB%D0%B5_%D0%B1%D0%B5%D0%BB%D0%B3%D1%96%D1%81%D1%96" title="Дирихле белгісі – 카자흐어" lang="kk" hreflang="kk" data-title="Дирихле белгісі" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Kryterium_Dirichleta_zbie%C5%BCno%C5%9Bci_szereg%C3%B3w_liczbowych" title="Kryterium Dirichleta zbieżności szeregów liczbowych – 폴란드어" lang="pl" hreflang="pl" data-title="Kryterium Dirichleta zbieżności szeregów liczbowych" data-language-autonym="Polski" data-language-local-name="폴란드어" 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/Teste_de_Dirichlet" title="Teste de Dirichlet – 포르투갈어" lang="pt" hreflang="pt" data-title="Teste de Dirichlet" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%B8%D0%B7%D0%BD%D0%B0%D0%BA_%D0%94%D0%B8%D1%80%D0%B8%D1%85%D0%BB%D0%B5" title="Признак Дирихле – 러시아어" lang="ru" hreflang="ru" data-title="Признак Дирихле" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Dirichlets_test" title="Dirichlets test – 스웨덴어" lang="sv" hreflang="sv" data-title="Dirichlets test" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dirichlet_testi" title="Dirichlet testi – 터키어" lang="tr" hreflang="tr" data-title="Dirichlet testi" data-language-autonym="Türkçe" data-language-local-name="터키어" 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%9E%D0%B7%D0%BD%D0%B0%D0%BA%D0%B0_%D0%94%D1%96%D1%80%D1%96%D1%85%D0%BB%D0%B5" title="Ознака Діріхле – 우크라이나어" lang="uk" hreflang="uk" data-title="Ознака Діріхле" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Dirixle_alomati_(testi)" title="Dirixle alomati (testi) – 우즈베크어" lang="uz" hreflang="uz" data-title="Dirixle alomati (testi)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%8B%84%E5%88%A9%E5%85%8B%E9%9B%B7%E5%88%A4%E5%88%AB%E6%B3%95" title="狄利克雷判别法 – 중국어" lang="zh" hreflang="zh" data-title="狄利克雷判别法" data-language-autonym="中文" data-language-local-name="중국어" 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 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href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311371"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">관련 문서 둘러보기</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="미적분학">미적분학</a></th></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99%EC%9D%98_%EA%B8%B0%EB%B3%B8%EC%A0%95%EB%A6%AC" class="mw-redirect" title="미적분학의 기본정리">미적분학의 기본정리</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B9%ED%95%9C" title="함수의 극한">함수의 극한</a></li> <li><a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 함수</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/%ED%8F%89%EA%B7%A0%EA%B0%92_%EC%A0%95%EB%A6%AC" title="평균값 정리">평균값 정리</a></li> <li><a href="/wiki/%EB%A1%A4%EC%9D%98_%EC%A0%95%EB%A6%AC" title="롤의 정리">롤의 정리</a></li></ul> </div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;display:block;margin-top:0.65em;"><span style="font-size:110%;"><a href="/wiki/%EB%AF%B8%EB%B6%84%ED%95%99" title="미분학">미분</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="padding:0.1em 0;line-height:1.2em;"> <ul><li><a href="/wiki/%EB%AF%B8%EB%B6%84" title="미분">미분</a> <ul><li><a href="/w/index.php?title=%EB%AF%B8%EB%B6%84%EC%9D%98_%EC%9D%BC%EB%B0%98%ED%99%94&action=edit&redlink=1" class="new" title="미분의 일반화 (없는 문서)">일반화</a></li> <li><a href="/wiki/%EB%AF%B8%EB%B6%84%EC%86%8C" title="미분소">무한소</a></li> <li><a href="/wiki/%EB%AF%B8%EB%B6%84_(%EC%A3%BC%EC%9A%94_%EB%B6%80%EB%B6%84)" title="미분 (주요 부분)">주요 부분</a></li> <li><a href="/wiki/%EC%A0%84%EB%AF%B8%EB%B6%84" title="전미분">전미분</a></li></ul></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> 개념</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist"> <ul><li><a href="/w/index.php?title=%EB%AF%B8%EB%B6%84_%ED%91%9C%EA%B8%B0%EB%B2%95&action=edit&redlink=1" class="new" title="미분 표기법 (없는 문서)">미분 표기법</a></li> <li><a href="/wiki/%EA%B3%A0%EA%B3%84_%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="고계 도함수">고계 도함수</a></li> <li><a href="/wiki/%EB%B3%80%EC%88%98_%EB%B3%80%ED%99%98" title="변수 변환">변수 변환</a></li> <li><a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EC%A0%95%EB%A6%AC" title="테일러 정리">테일러 정리</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/%EB%AF%B8%EB%B6%84%ED%91%9C" class="mw-redirect" title="미분표">법칙과 항등식</a></th></tr><tr><td class="sidebar-content hlist"> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%A9_%EA%B7%9C%EC%B9%99" title="합 규칙">합 규칙</a></li> <li><a href="/wiki/%EA%B3%B1_%EA%B7%9C%EC%B9%99" title="곱 규칙">곱 규칙</a></li> <li><a href="/wiki/%EB%AA%AB_%EA%B7%9C%EC%B9%99" title="몫 규칙">몫 규칙</a></li> <li><a href="/wiki/%EB%A9%B1_%EA%B7%9C%EC%B9%99" title="멱 규칙">멱 규칙</a></li> <li><a href="/wiki/%EC%97%B0%EC%87%84_%EB%B2%95%EC%B9%99" title="연쇄 법칙">연쇄 법칙</a></li> <li><a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98%EC%9D%98_%EB%AF%B8%EB%B6%84" class="mw-redirect" title="역함수의 미분">역함수의 미분</a></li> <li><a href="/wiki/%EC%9D%8C%ED%95%A8%EC%88%98%EC%9D%98_%EB%AF%B8%EB%B6%84" class="mw-redirect" title="음함수의 미분">음함수의 미분</a></li></ul> </div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EC%A0%81%EB%B6%84%ED%91%9C" title="적분표">적분표</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B6%80%EC%A0%95%EC%A0%81%EB%B6%84" title="부정적분">부정적분</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a> (<a href="/wiki/%EC%9D%B4%EC%83%81%EC%A0%81%EB%B6%84" class="mw-redirect" title="이상적분">이상적분</a>)</li> <li><a href="/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%81%EB%B6%84" title="리만 적분">리만 적분</a></li> <li><a href="/wiki/%EB%A5%B4%EB%B2%A0%EA%B7%B8_%EC%A0%81%EB%B6%84" title="르베그 적분">르베그 적분</a></li> <li><a href="/wiki/%EA%B2%BD%EB%A1%9C%EC%A0%81%EB%B6%84%EB%B2%95" title="경로적분법">경로적분</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 적분법</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B6%80%EB%B6%84%EC%A0%81%EB%B6%84" class="mw-redirect" title="부분적분">부분적분</a></li> <li><a href="/w/index.php?title=%EB%94%94%EC%8A%A4%ED%81%AC_%EB%B0%A9%EB%B2%95&action=edit&redlink=1" class="new" title="디스크 방법 (없는 문서)">디스크 방법</a></li> <li><a href="/wiki/%EC%9B%90%ED%86%B5%EC%85%B8_%EB%B0%A9%EB%B2%95" title="원통셸 방법">원통셸 방법</a></li> <li><a href="/wiki/%EC%B9%98%ED%99%98%EC%A0%81%EB%B6%84" class="mw-redirect" title="치환적분">치환적분</a> (<a href="/wiki/%EC%82%BC%EA%B0%81_%EC%B9%98%ED%99%98" title="삼각 치환">삼각 치환</a>)</li> <li><a href="/wiki/%EB%B6%80%EB%B6%84%EB%B6%84%EC%88%98" title="부분분수">부분분수 적분법</a></li> <li><a href="/w/index.php?title=%EC%A0%81%EB%B6%84_%EC%88%9C%EC%84%9C&action=edit&redlink=1" class="new" title="적분 순서 (없는 문서)">적분 순서</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84%EC%9D%98_%EC%A0%90%ED%99%94%EC%8B%9D" title="적분의 점화식">적분의 점화식</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a>과 <a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EA%B8%B0%ED%95%98%EA%B8%89%EC%88%98" class="mw-redirect" title="기하급수">기하급수</a> (<a href="/wiki/%EC%82%B0%EC%88%A0-%EA%B8%B0%ED%95%98_%EC%88%98%EC%97%B4" title="산술-기하 수열">산술-기하 수열</a>)</li> <li><a href="/wiki/%EC%A1%B0%ED%99%94%EA%B8%89%EC%88%98" title="조화급수">조화급수</a></li> <li><a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98" title="교대급수">교대급수</a></li> <li><a href="/wiki/%EB%A9%B1%EA%B8%89%EC%88%98" title="멱급수">멱급수</a></li> <li><a href="/wiki/%EC%9D%B4%ED%95%AD%EA%B8%89%EC%88%98" class="mw-redirect" title="이항급수">이항급수</a></li> <li><a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EA%B8%89%EC%88%98" title="테일러 급수">테일러 급수</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/%EC%88%98%EB%A0%B4%ED%8C%90%EC%A0%95%EB%B2%95" title="수렴판정법">수렴판정법</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EC%9D%BC%EB%B0%98%ED%95%AD_%ED%8C%90%EC%A0%95%EB%B2%95" title="일반항 판정법">일반항 판정법</a></li> <li><a href="/wiki/%EB%B9%84%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="비판정법">비판정법</a></li> <li><a href="/wiki/%EA%B7%BC%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="근판정법">근판정법</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="적분판정법">적분판정법</a></li> <li><a href="/wiki/%EB%B9%84%EA%B5%90%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="비교판정법">비교판정법</a></li> <li><a href="/wiki/%EA%B7%B9%ED%95%9C%EB%B9%84%EA%B5%90%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="극한비교판정법">극한비교판정법</a></li> <li><a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="교대급수판정법">교대급수판정법</a></li> <li><a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%9D%91%EC%A7%91%ED%8C%90%EC%A0%95%EB%B2%95" title="코시 응집판정법">코시 응집판정법</a></li> <li><a class="mw-selflink selflink">디리클레 판정법</a></li> <li><a href="/wiki/%EC%95%84%EB%B2%A8_%ED%8C%90%EC%A0%95%EB%B2%95" title="아벨 판정법">아벨 판정법</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="벡터 미적분학">벡터 미적분학</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)" title="기울기 (벡터)">기울기</a></li> <li><a href="/wiki/%EB%B0%9C%EC%82%B0_(%EB%B2%A1%ED%84%B0)" title="발산 (벡터)">발산</a></li> <li><a href="/wiki/%ED%9A%8C%EC%A0%84_(%EB%B2%A1%ED%84%B0)" title="회전 (벡터)">회전</a></li> <li><a href="/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EC%97%B0%EC%82%B0%EC%9E%90" title="라플라스 연산자">라플라시안</a></li> <li><a href="/wiki/%EB%B0%A9%ED%96%A5%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="방향도함수">방향도함수</a></li> <li><a href="/w/index.php?title=%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%91%9C&action=edit&redlink=1" class="new" title="벡터 미적분표 (없는 문서)">벡터 미적분표</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정리</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B0%9C%EC%82%B0%EC%A0%95%EB%A6%AC" class="mw-redirect" title="발산정리">발산정리</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0%EC%A0%95%EB%A6%AC&action=edit&redlink=1" class="new" title="기울기정리 (없는 문서)">기울기정리</a></li> <li><a href="/wiki/%EA%B7%B8%EB%A6%B0_%EC%A0%95%EB%A6%AC" title="그린 정리">그린 정리</a></li> <li><a href="/wiki/%EC%BC%88%EB%B9%88-%EC%8A%A4%ED%86%A0%ED%81%AC%EC%8A%A4_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="켈빈-스토크스 정리">켈빈-스토크스 정리</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EB%8B%A4%EB%B3%80%EC%88%98_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" class="mw-redirect" title="다변수 미적분학">다변수 미적분학</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading"> 형식</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/w/index.php?title=%ED%96%89%EB%A0%AC_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99&action=edit&redlink=1" class="new" title="행렬 미적분학 (없는 문서)">행렬</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="텐서 미적분학">텐서</a></li> <li><a href="/wiki/%EC%99%B8%EB%AF%B8%EB%B6%84" class="mw-redirect" title="외미분">외미분</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%ED%95%98_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99&action=edit&redlink=1" class="new" title="기하 미적분학 (없는 문서)">기하</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%ED%8E%B8%EB%AF%B8%EB%B6%84" title="편미분">편미분</a></li> <li><a href="/wiki/%EC%A4%91%EC%A0%81%EB%B6%84" title="중적분">중적분</a></li> <li><a href="/wiki/%EC%84%A0%EC%A0%81%EB%B6%84" title="선적분">선적분</a></li> <li><a href="/wiki/%EB%A9%B4%EC%A0%81%EB%B6%84" title="면적분">면적분</a></li> <li><a href="/wiki/%EC%82%BC%EC%A4%91%EC%A0%81%EB%B6%84" class="mw-redirect" title="삼중적분">삼중적분</a></li> <li><a href="/wiki/%EC%95%BC%EC%BD%94%EB%B9%84_%ED%96%89%EB%A0%AC" title="야코비 행렬">야코비안</a></li> <li><a href="/wiki/%ED%97%A4%EC%84%B8_%ED%96%89%EB%A0%AC" title="헤세 행렬">헤세 행렬</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;">특수한 경우</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EB%B6%84%EC%88%98%EA%B3%84_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="분수계 미적분학">분수계 미적분학</a></li> <li><a href="/w/index.php?title=%EB%A7%90%EB%A6%AC%EC%95%84%EB%B9%88_%EB%AF%B8%EC%A0%81%EB%B6%84&action=edit&redlink=1" class="new" title="말리아빈 미적분 (없는 문서)">말리아빈 미적분</a></li> <li><a href="/wiki/%ED%99%95%EB%A5%A0%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="확률미적분학">확률미적분학</a></li> <li><a href="/wiki/%EB%B3%80%EB%B6%84%EB%B2%95" title="변분법">변분법</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><style data-mw-deduplicate="TemplateStyles:r34311309">.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 .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-보기"><a href="/wiki/%ED%8B%80:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="틀:미적분학"><abbr title="이 틀을 보기">v</abbr></a></li><li class="nv-토론"><a href="/wiki/%ED%8B%80%ED%86%A0%EB%A1%A0:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="틀토론:미적분학"><abbr title="이 틀에 관해 토론하기">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="특수:문서편집/틀:미적분학"><abbr title="이 틀을 편집하기">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><a href="/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="미적분학">미적분학</a>에서 <b>디리클레 판정법</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">Dirichlet's test</span>)은 <a href="/wiki/%EC%8B%A4%EC%88%98_%ED%95%AD_%EA%B8%89%EC%88%98" class="mw-redirect" title="실수 항 급수">실수 항 급수</a>의 <a href="/wiki/%EC%88%98%EB%A0%B4_%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="수렴 판정법">수렴 판정법</a>의 하나다. 이에 따르면, <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%A7%91%ED%95%A9" title="유계 집합">유계</a> <a href="/wiki/%EB%B6%80%EB%B6%84%ED%95%A9" class="mw-redirect" title="부분합">부분합</a>을 갖는 급수에 0으로 수렴하는 <a href="/wiki/%EB%8B%A8%EC%A1%B0%EC%88%98%EC%97%B4" class="mw-redirect" title="단조수열">단조수열</a>을 계수로서 곱한 급수는 수렴한다. <a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98_%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="교대급수 판정법">교대급수 판정법</a>을 일반화한다. 디리클레 판정법의 표준적인 증명은 유한합의 <a href="/wiki/%EC%95%84%EB%B2%A8_%EB%B3%80%ED%99%98" title="아벨 변환">아벨 변환</a>을 사용한다. <a href="/wiki/%EC%9D%B4%EC%83%81_%EC%A0%81%EB%B6%84" title="이상 적분">이상 적분</a>에 대한 디리클레 판정법은 <a href="/wiki/%EC%A0%9C2_%EC%A0%81%EB%B6%84_%ED%8F%89%EA%B7%A0%EA%B0%92_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="제2 적분 평균값 정리">제2 적분 평균값 정리</a>를 통하여 보일 수 있는데, 이에 대한 증명은 <a href="/wiki/%EC%95%84%EB%B2%A8_%EB%B3%80%ED%99%98" title="아벨 변환">아벨 변환</a>을 필요로 한다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="실수_항_급수"><span id=".EC.8B.A4.EC.88.98_.ED.95.AD_.EA.B8.89.EC.88.98"></span>실수 항 급수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=2" title="부분 편집: 실수 항 급수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>두 <a href="/wiki/%EC%8B%A4%EC%88%98_%EC%88%98%EC%97%B4" class="mw-redirect" title="실수 수열">실수 수열</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f97380c15f3601c311463d76e6a03798a360e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.009ex;" alt="{\displaystyle (a_{n})_{n=0}^{\infty }}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae76f4351995d2ad292ae03d2eb778b7a7f68b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.344ex; height:3.009ex;" alt="{\displaystyle (b_{n})_{n=0}^{\infty }}"></span>이 다음 세 조건을 만족시킨다고 하자. </p> <ul><li>급수 <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 }a_{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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0af34647e168beb46e51ff2e4547712cf3f9d4ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.19ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a_{n}}"></span>의 <a href="/wiki/%EB%B6%80%EB%B6%84%ED%95%A9" class="mw-redirect" title="부분합">부분합</a>은 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%88%98%EC%97%B4" class="mw-redirect" title="유계 수열">유계 수열</a>이다. 즉, <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 \sup _{n\in \mathbb {N} }\left|{\sum _{k=0}^{n}a_{k}}\right|<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{n\in \mathbb {N} }\left|{\sum _{k=0}^{n}a_{k}}\right|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da589b379bb5b388a4da9ce93eb69d91329cad14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.665ex; height:7.509ex;" alt="{\displaystyle \sup _{n\in \mathbb {N} }\left|{\sum _{k=0}^{n}a_{k}}\right|<\infty }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae76f4351995d2ad292ae03d2eb778b7a7f68b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.344ex; height:3.009ex;" alt="{\displaystyle (b_{n})_{n=0}^{\infty }}"></span>은 <a href="/wiki/%EB%8B%A8%EC%A1%B0%EC%88%98%EC%97%B4" class="mw-redirect" title="단조수열">단조수열</a>이다. 즉, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}\geq b_{1}\geq b_{2}\geq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≥</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}\geq b_{1}\geq b_{2}\geq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa27244c0ccde124a9f948870e1a929b62f7670b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.174ex; height:2.509ex;" alt="{\displaystyle b_{0}\geq b_{1}\geq b_{2}\geq \cdots }"></span>이거나 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}\leq b_{1}\leq b_{2}\leq \cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{0}\leq b_{1}\leq b_{2}\leq \cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25c3008d14cb2c6ea6cbdc4ab2361657133e0acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.174ex; height:2.509ex;" alt="{\displaystyle b_{0}\leq b_{1}\leq b_{2}\leq \cdots }"></span></li> <li><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 }b_{n}=0}"> <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> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }b_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3162a00e0fc9d70a048f24a8d4eb66a193ca0d56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.137ex; height:3.676ex;" alt="{\displaystyle \lim _{n\to \infty }b_{n}=0}"></span></li></ul> <p><b>디리클레 판정법</b>에 따르면, 급수 </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 }a_{n}b_{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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a_{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956a9319059c4cb6e3e9f246cb7f1018687c600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.406ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a_{n}b_{n}}"></span></dd></dl> <p>는 <a href="/wiki/%EC%88%98%EB%A0%B4" class="mw-redirect" title="수렴">수렴</a>한다.<sup id="cite_ref-김락중_1-0" class="reference"><a href="#cite_note-김락중-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:182</sup></span><sup id="cite_ref-Knopp_2-0" class="reference"><a href="#cite_note-Knopp-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:315, °2</sup></span> </p> <style data-mw-deduplicate="TemplateStyles:r26858958">.mw-parser-output div.proof{border:1px solid #aaaaaa;background-color:#f9f9f9;padding:5px;font-size:95%;min-width:50%}.mw-parser-output div.proof,.mw-parser-output div.prooftitle,.mw-parser-output div.proofcontent{overflow:auto}.mw-parser-output div.prooftitle span.prooftitletext{font-weight:bold}.mw-parser-output div.proofcontent{margin-top:-0.5em;min-height:0.5em}</style><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명:</span><sup id="cite_ref-김락중_1-1" class="reference"><a href="#cite_note-김락중-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> </div> <div class="proofcontent mw-collapsible-content"> <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 S_{n}=\sum _{k=0}^{n}a_{k}\qquad (\forall n\in \mathbb {N} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}=\sum _{k=0}^{n}a_{k}\qquad (\forall n\in \mathbb {N} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cda9d2f6c63233f191123ece79164f201ca8b980" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.463ex; height:7.009ex;" alt="{\displaystyle S_{n}=\sum _{k=0}^{n}a_{k}\qquad (\forall n\in \mathbb {N} )}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=1+\sup _{n\in \mathbb {N} }|S_{n}|\in \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=1+\sup _{n\in \mathbb {N} }|S_{n}|\in \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d32d8a597fcd85cc1e10f6e36b4727ee00d3448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.399ex; height:4.676ex;" alt="{\displaystyle M=1+\sup _{n\in \mathbb {N} }|S_{n}|\in \mathbb {R} ^{+}}"></span></dd></dl> <p>라고 하자. 임의의 <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>에 대하여, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |b_{n}|<{\frac {\epsilon }{6M}}\qquad (\forall n>N(\epsilon ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <mn>6</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>n</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |b_{n}|<{\frac {\epsilon }{6M}}\qquad (\forall n>N(\epsilon ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/457a0e47cdc0f5527856f7d7ec08c97b3a5b5808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.106ex; height:4.843ex;" alt="{\displaystyle |b_{n}|<{\frac {\epsilon }{6M}}\qquad (\forall n>N(\epsilon ))}"></span></dd></dl> <p>인 자연수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\epsilon )\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\epsilon )\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616334231c5269acffa23d9df6d3f213884f7ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.336ex; height:2.843ex;" alt="{\displaystyle N(\epsilon )\in \mathbb {N} }"></span>이 존재한다. <a href="/wiki/%EC%95%84%EB%B2%A8_%EB%B3%80%ED%99%98" title="아벨 변환">아벨 변환</a>에 의하여, 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/caa09a0f17ee55e908afd9d8ef52671056a1c798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.31ex; height:2.843ex;" alt="{\displaystyle n\geq N(\epsilon )}"></span> 및 임의의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {Z} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bffb7c9f0c6cfe4004af33ab51dd67c76cc73a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.161ex; height:2.843ex;" alt="{\displaystyle p\in \mathbb {Z} ^{+}}"></span>에 대하여, 다음이 성립한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|\sum _{k=n+1}^{n+p}a_{k}b_{k}\right|&=\left|b_{n+p}(S_{n+p}-S_{n})+\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})(S_{k}-S_{n})\right|\\&\leq |b_{n+p}|(|S_{n+p}|+|S_{n}|)+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|(|S_{k}|+|S_{n}|)\\&\leq 2M\left(|b_{n+p}|+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|\right)\\&=2M\left(|b_{n+p}|+\left|\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})\right|\right)\\&=2M(|b_{n+p}|+|b_{n+1}-b_{n+p}|)\\&\leq 2M(2|b_{n+p}|+|b_{n+1}|)\\&<2M\cdot 3\cdot {\frac {\epsilon }{6M}}\\&=\epsilon \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> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mn>2</mn> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>M</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>M</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mn>2</mn> <mi>M</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo><</mo> <mn>2</mn> <mi>M</mi> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <mn>6</mn> <mi>M</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ϵ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|\sum _{k=n+1}^{n+p}a_{k}b_{k}\right|&=\left|b_{n+p}(S_{n+p}-S_{n})+\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})(S_{k}-S_{n})\right|\\&\leq |b_{n+p}|(|S_{n+p}|+|S_{n}|)+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|(|S_{k}|+|S_{n}|)\\&\leq 2M\left(|b_{n+p}|+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|\right)\\&=2M\left(|b_{n+p}|+\left|\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})\right|\right)\\&=2M(|b_{n+p}|+|b_{n+1}-b_{n+p}|)\\&\leq 2M(2|b_{n+p}|+|b_{n+1}|)\\&<2M\cdot 3\cdot {\frac {\epsilon }{6M}}\\&=\epsilon \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f633d86db1b6cc75a57da8980a73a0406188414c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -22.005ex; width:66.652ex; height:45.176ex;" alt="{\displaystyle {\begin{aligned}\left|\sum _{k=n+1}^{n+p}a_{k}b_{k}\right|&=\left|b_{n+p}(S_{n+p}-S_{n})+\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})(S_{k}-S_{n})\right|\\&\leq |b_{n+p}|(|S_{n+p}|+|S_{n}|)+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|(|S_{k}|+|S_{n}|)\\&\leq 2M\left(|b_{n+p}|+\sum _{k=n+1}^{n+p-1}|b_{k}-b_{k+1}|\right)\\&=2M\left(|b_{n+p}|+\left|\sum _{k=n+1}^{n+p-1}(b_{k}-b_{k+1})\right|\right)\\&=2M(|b_{n+p}|+|b_{n+1}-b_{n+p}|)\\&\leq 2M(2|b_{n+p}|+|b_{n+1}|)\\&<2M\cdot 3\cdot {\frac {\epsilon }{6M}}\\&=\epsilon \end{aligned}}}"></span></dd></dl> <p>즉, 급수 </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 }a_{n}b_{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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a_{n}b_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956a9319059c4cb6e3e9f246cb7f1018687c600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.406ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }a_{n}b_{n}}"></span></dd></dl> <p>의 <a href="/wiki/%EB%B6%80%EB%B6%84%ED%95%A9" class="mw-redirect" title="부분합">부분합</a>은 <a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%88%98%EC%97%B4" class="mw-redirect" title="코시 수열">코시 수열</a>이다. 따라서 이 급수는 수렴한다. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="이상_적분"><span id=".EC.9D.B4.EC.83.81_.EC.A0.81.EB.B6.84"></span>이상 적분</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=3" title="부분 편집: 이상 적분"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>두 <a href="/wiki/%EC%8B%A4%EC%88%98_%EA%B0%92_%ED%95%A8%EC%88%98" class="mw-redirect" title="실수 값 함수">실수 값 함수</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon [a,\infty )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g\colon [a,\infty )\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1cc96fd142cc7aed74423534f75102c99aa6d19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.894ex; height:2.843ex;" alt="{\displaystyle f,g\colon [a,\infty )\to \mathbb {R} }"></span>가 다음 세 조건을 만족시킨다고 하자. </p> <ul><li><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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>는 임의의 <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 [a,b]\subseteq [a,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>⊆</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]\subseteq [a,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c274f2ee602fdde02ddcde46fefd86f610583828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.792ex; height:2.843ex;" alt="{\displaystyle [a,b]\subseteq [a,\infty )}"></span>에서 <a href="/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%81%EB%B6%84" title="리만 적분">리만 적분</a> 가능하며, 또한 <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 \sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fcb5015cb35c45d721835683293ac2cbb2e51e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.6ex; height:6.509ex;" alt="{\displaystyle \sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|<\infty }"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>는 <a href="/wiki/%EB%8B%A8%EC%A1%B0%ED%95%A8%EC%88%98" title="단조함수">단조함수</a>이다. (특히, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>는 임의의 <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 [a,b]\subseteq [a,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>⊆</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]\subseteq [a,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c274f2ee602fdde02ddcde46fefd86f610583828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.792ex; height:2.843ex;" alt="{\displaystyle [a,b]\subseteq [a,\infty )}"></span>에서 <a href="/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%81%EB%B6%84" title="리만 적분">리만 적분</a> 가능하다.)</li> <li><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 _{x\to \infty }g(x)=0}"> <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>x</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{x\to \infty }g(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f4598ed48a427cbbc9c521dbb252aeac26267ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.13ex; height:3.843ex;" alt="{\displaystyle \lim _{x\to \infty }g(x)=0}"></span></li></ul> <p>그렇다면, <a href="/wiki/%EC%9D%B4%EC%83%81_%EC%A0%81%EB%B6%84" title="이상 적분">이상 적분</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024bf4e84ec9f7c42675dd3afc4eb4df3dcf4ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.332ex; height:5.843ex;" alt="{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}"></span></dd></dl> <p>는 수렴한다. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26858958"><div class="proof mw-collapsible mw-collapsed"> <div class="prooftitle"> <p><span class="prooftitletext">증명:</span> </p> </div> <div class="proofcontent mw-collapsible-content"> <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 M=1+\sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|\in \mathbb {R} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>1</mn> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=1+\sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|\in \mathbb {R} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6901549ba98da3d3af092a1c7c772f50ee8fb4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.751ex; height:6.509ex;" alt="{\displaystyle M=1+\sup _{x\in [a,\infty )}\left|\int _{a}^{x}f(t)\,dt\right|\in \mathbb {R} ^{+}}"></span></dd></dl> <p>이라고 하자. 임의의 <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 \epsilon >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568095ad3924314374a5ab68fae17343661f2a71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}"></span>에 대하여, </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 |g(x)|<{\frac {\epsilon }{4M}}\qquad (\forall x>N(\epsilon ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |g(x)|<{\frac {\epsilon }{4M}}\qquad (\forall x>N(\epsilon ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a69735c5509cd207415b3eb225496b12674b13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.08ex; height:4.676ex;" alt="{\displaystyle |g(x)|<{\frac {\epsilon }{4M}}\qquad (\forall x>N(\epsilon ))}"></span></dd></dl> <p>인 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(\epsilon )>a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(\epsilon )>a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5097822b8bebe52a958791402cb3cb505054e67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle N(\epsilon )>a}"></span>가 존재한다. <a href="/wiki/%EC%A0%9C2_%EC%A0%81%EB%B6%84_%ED%8F%89%EA%B7%A0%EA%B0%92_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="제2 적분 평균값 정리">제2 적분 평균값 정리</a>에 따라, 임의의 <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 y>x>N(\epsilon )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>></mo> <mi>x</mi> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y>x>N(\epsilon )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1b9dd46fc6482492c17984bd2a861ce4af2d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.499ex; height:2.843ex;" alt="{\displaystyle y>x>N(\epsilon )}"></span>에 대하여, 어떤 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(x,y)\in [x,y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(x,y)\in [x,y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc8b0f64c4246d7823ec22c20ae33f62a63df71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.989ex; height:2.843ex;" alt="{\displaystyle c(x,y)\in [x,y]}"></span>가 존재하며, 다음이 성립한다. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\left|\int _{x}^{y}f(t)g(t)\,dt\right|&=\left|g(x)\int _{x}^{c(x,y)}f(t)\,dt+g(y)\int _{c(x,y)}^{y}f(t)\,dt\right|\\&\leq {\frac {\epsilon }{4M}}\left(\left|\int _{a}^{c(x,y)}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right|+\left|\int _{a}^{y}f(t)\,dt-\int _{a}^{c(x,y)}f(t)\,dt\right|\right)\\&\leq {\frac {\epsilon }{4M}}(2M+2M)\\&=\epsilon \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> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> <mo>+</mo> <mrow> <mo>|</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ϵ</mi> <mrow> <mn>4</mn> <mi>M</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>M</mi> <mo>+</mo> <mn>2</mn> <mi>M</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ϵ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|\int _{x}^{y}f(t)g(t)\,dt\right|&=\left|g(x)\int _{x}^{c(x,y)}f(t)\,dt+g(y)\int _{c(x,y)}^{y}f(t)\,dt\right|\\&\leq {\frac {\epsilon }{4M}}\left(\left|\int _{a}^{c(x,y)}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right|+\left|\int _{a}^{y}f(t)\,dt-\int _{a}^{c(x,y)}f(t)\,dt\right|\right)\\&\leq {\frac {\epsilon }{4M}}(2M+2M)\\&=\epsilon \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5632ef3e0e3874f9fd8985773e3491a24b915dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:86.781ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}\left|\int _{x}^{y}f(t)g(t)\,dt\right|&=\left|g(x)\int _{x}^{c(x,y)}f(t)\,dt+g(y)\int _{c(x,y)}^{y}f(t)\,dt\right|\\&\leq {\frac {\epsilon }{4M}}\left(\left|\int _{a}^{c(x,y)}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right|+\left|\int _{a}^{y}f(t)\,dt-\int _{a}^{c(x,y)}f(t)\,dt\right|\right)\\&\leq {\frac {\epsilon }{4M}}(2M+2M)\\&=\epsilon \end{aligned}}}"></span></dd></dl> <p>따라서, <a href="/wiki/%EC%9D%B4%EC%83%81_%EC%A0%81%EB%B6%84" title="이상 적분">이상 적분</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/024bf4e84ec9f7c42675dd3afc4eb4df3dcf4ee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.332ex; height:5.843ex;" alt="{\displaystyle \int _{a}^{\infty }f(x)g(x)\,dx}"></span></dd></dl> <p>은 수렴한다. </p> </div></div> <div class="mw-heading mw-heading3"><h3 id="균등_수렴"><span id=".EA.B7.A0.EB.93.B1_.EC.88.98.EB.A0.B4"></span>균등 수렴</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=4" title="부분 편집: 균등 수렴"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>집합 <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 및 두 함수열 <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 (f_{n},g_{n}\colon X\to \mathbb {R} )_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{n},g_{n}\colon X\to \mathbb {R} )_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fa69a96da3e60fe2e79de3f6684ce643d0bad9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.154ex; height:3.009ex;" alt="{\displaystyle (f_{n},g_{n}\colon X\to \mathbb {R} )_{n=0}^{\infty }}"></span>이 다음 세 조건을 만족시킨다고 하자. </p> <ul><li>함수 항 급수 <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 }f_{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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac338d9c025aa84b3ae88e6b02bf49d1a27f9915" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:6.1ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }f_{n}}"></span>의 부분합은 <a href="/wiki/%EA%B7%A0%EB%93%B1_%EC%9C%A0%EA%B3%84_%ED%95%A8%EC%88%98%EC%97%B4" class="mw-redirect" title="균등 유계 함수열">균등 유계 함수열</a>이다. 즉, <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 \sup _{n\in \mathbb {N} }\sup _{x\in X}\left|\sum _{i=0}^{n}f_{i}(x)\right|<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{n\in \mathbb {N} }\sup _{x\in X}\left|\sum _{i=0}^{n}f_{i}(x)\right|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2a3e45ec96068e72287c8c5bf03ff4c1d97d0ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.313ex; height:7.176ex;" alt="{\displaystyle \sup _{n\in \mathbb {N} }\sup _{x\in X}\left|\sum _{i=0}^{n}f_{i}(x)\right|<\infty }"></span></li> <li>임의의 <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 x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span>에 대하여, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{n}(x))_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{n}(x))_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed204b71262bed4e09ecebf612afb4e48ca9fcec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.595ex; height:3.009ex;" alt="{\displaystyle (g_{n}(x))_{n=0}^{\infty }}"></span>은 <a href="/wiki/%EB%8B%A8%EC%A1%B0%EC%88%98%EC%97%B4" class="mw-redirect" title="단조수열">단조수열</a>이다.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfa3fa4b4b16da96f4bb33c7a922db507cf873f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.456ex; height:3.009ex;" alt="{\displaystyle (g_{n})_{n=0}^{\infty }}"></span>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\colon X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\colon X\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92224c47ad219fe6b3420a09023072eded3cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.469ex; height:2.176ex;" alt="{\displaystyle 0\colon X\to \mathbb {R} }"></span>로 <a href="/wiki/%EA%B7%A0%EB%93%B1_%EC%88%98%EB%A0%B4" title="균등 수렴">균등 수렴</a>한다.</li></ul> <p>그렇다면, 급수 </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 }f_{n}g_{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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{n}g_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90e4f1958d985a30bd81155c6d3dd4506f8cb180" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:8.427ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }f_{n}g_{n}}"></span></dd></dl> <p>는 <a href="/wiki/%EA%B7%A0%EB%93%B1_%EC%88%98%EB%A0%B4" title="균등 수렴">균등 수렴</a>한다. 이에 대한 증명은 실수 항 급수에 대한 디리클레 판정법의 증명과 유사하다. <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>가 <a href="/wiki/%ED%95%9C%EC%9B%90%EC%86%8C_%EC%A7%91%ED%95%A9" title="한원소 집합">한원소 집합</a>인 경우, 이는 단순히 실수 항 급수에 대한 디리클레 판정법이다. </p> <div class="mw-heading mw-heading2"><h2 id="예"><span id=".EC.98.88"></span>예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=5" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="교대급수"><span id=".EA.B5.90.EB.8C.80.EA.B8.89.EC.88.98"></span>교대급수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=6" title="부분 편집: 교대급수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.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}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98" title="교대급수">교대급수</a>입니다.</div> <p>임의의 0으로 수렴하는 <a href="/wiki/%EB%8B%A8%EC%A1%B0%EC%88%98%EC%97%B4" class="mw-redirect" title="단조수열">단조수열</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n=0}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</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> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n=0}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f97380c15f3601c311463d76e6a03798a360e4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.009ex;" alt="{\displaystyle (a_{n})_{n=0}^{\infty }}"></span>에 대하여, <a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98" title="교대급수">교대급수</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{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> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfe82b2e7b054f2c7ea25d5b4439e43cc62e41c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.802ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}}"></span></dd></dl> <p>는 수렴한다 (<a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98_%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="교대급수 판정법">교대급수 판정법</a>). 이는 급수 </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>의 부분합 </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 _{i=0}^{n}(-1)^{i}={\begin{cases}1&n\in 2\mathbb {Z} \\0&n\in 2\mathbb {Z} +1\end{cases}}\qquad (n\in \mathbb {N} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</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> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>n</mi> <mo>∈</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <mo>∈</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}(-1)^{i}={\begin{cases}1&n\in 2\mathbb {Z} \\0&n\in 2\mathbb {Z} +1\end{cases}}\qquad (n\in \mathbb {N} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad5b13da8dcd76e9c1d9d03542cfa179652d86b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.332ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}(-1)^{i}={\begin{cases}1&n\in 2\mathbb {Z} \\0&n\in 2\mathbb {Z} +1\end{cases}}\qquad (n\in \mathbb {N} )}"></span></dd></dl> <p>이 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%88%98%EC%97%B4" class="mw-redirect" title="유계 수열">유계 수열</a>이기 때문이다. </p> <div class="mw-heading mw-heading3"><h3 id="삼각_급수"><span id=".EC.82.BC.EA.B0.81_.EA.B8.89.EC.88.98"></span>삼각 급수</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=7" title="부분 편집: 삼각 급수"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> <a href="/wiki/%ED%91%B8%EB%A6%AC%EC%97%90_%EA%B8%89%EC%88%98" title="푸리에 급수">푸리에 급수</a> 문서를 참고하십시오.</div> <p>마찬가자로, 0으로 수렴하는 <a href="/wiki/%EB%8B%A8%EC%A1%B0%EC%88%98%EC%97%B4" class="mw-redirect" title="단조수열">단조수열</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{n})_{n=1}^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{n})_{n=1}^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/548835a1d47d9829b180e1deda337ebf1bade908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.577ex; height:3.009ex;" alt="{\displaystyle (a_{n})_{n=1}^{\infty }}"></span> 및 <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 x\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c6d458566aec47a7259762034790c8981aefab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.848ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {R} }"></span>에 대하여, </p> <ul><li>급수 <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=1}^{\infty }a_{n}\sin nx}"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }a_{n}\sin nx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573e9802dd987a2a7db938e3a7f0605bfc87ac0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.545ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }a_{n}\sin nx}"></span>는 수렴한다.</li> <li>만약 <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 x\not \in 2\pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in 2\pi \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a07979e321b6cc159d36c64ba8f4a0647efaa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.215ex; height:2.676ex;" alt="{\displaystyle x\not \in 2\pi \mathbb {Z} }"></span>라면, 급수 <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=1}^{\infty }a_{n}\cos nx}"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }a_{n}\cos nx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54532c7e6cbc22d253df1c48d74202b82b9396ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.8ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }a_{n}\cos nx}"></span>는 수렴한다.</li></ul> <p>이는 </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 _{i=1}^{n}\sin ix={\begin{cases}0&x\in \pi \mathbb {Z} \\(\cos(x/2)-\cos((n+1/2)x))/(2\sin(x/2))&x\not \in \pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>sin</mi> <mo>⁡</mo> <mi>i</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>x</mi> <mo>∉</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\sin ix={\begin{cases}0&x\in \pi \mathbb {Z} \\(\cos(x/2)-\cos((n+1/2)x))/(2\sin(x/2))&x\not \in \pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaaab64de293a81fd06cc6ec8f8e02e0f5a29084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:79.684ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}\sin ix={\begin{cases}0&x\in \pi \mathbb {Z} \\(\cos(x/2)-\cos((n+1/2)x))/(2\sin(x/2))&x\not \in \pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}"></span></dd></dl> <p>가 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%88%98%EC%97%B4" class="mw-redirect" title="유계 수열">유계 수열</a>이며, </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 _{i=1}^{n}\cos ix={\begin{cases}n&x\in 2\pi \mathbb {Z} \\(\sin((n+1/2)x)-\sin(x/2))/(2\sin(x/2))&x\not \in 2\pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>cos</mi> <mo>⁡</mo> <mi>i</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi>x</mi> <mo>∈</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>x</mi> <mo>∉</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}\cos ix={\begin{cases}n&x\in 2\pi \mathbb {Z} \\(\sin((n+1/2)x)-\sin(x/2))/(2\sin(x/2))&x\not \in 2\pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1802ea930af9d32114a83ca893dddfe6a7e3bad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.591ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}\cos ix={\begin{cases}n&x\in 2\pi \mathbb {Z} \\(\sin((n+1/2)x)-\sin(x/2))/(2\sin(x/2))&x\not \in 2\pi \mathbb {Z} \end{cases}}\qquad (n\in \mathbb {Z} ^{+})}"></span></dd></dl> <p>가 <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 x\not \in 2\pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∉</mo> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\not \in 2\pi \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a07979e321b6cc159d36c64ba8f4a0647efaa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.215ex; height:2.676ex;" alt="{\displaystyle x\not \in 2\pi \mathbb {Z} }"></span>일 때 <a href="/wiki/%EC%9C%A0%EA%B3%84_%EC%88%98%EC%97%B4" class="mw-redirect" title="유계 수열">유계 수열</a>이기 때문이다. 또한, 다음 두 조건이 서로 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li><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=1}^{\infty }|a_{n}\sin nx|<\infty }"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }|a_{n}\sin nx|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a887340099fd79749edf24fceb464dce295eba2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.26ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }|a_{n}\sin nx|<\infty }"></span></li> <li><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 x\in \pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \pi \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae0fe6aaea607b046092673617839ab7e6eb69c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.053ex; height:2.176ex;" alt="{\displaystyle x\in \pi \mathbb {Z} }"></span>이거나, <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=1}^{\infty }|a_{n}|<\infty }"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }|a_{n}|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d363ec23ac87f20c95a403ec41b52f1ef63799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.906ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }|a_{n}|<\infty }"></span></li></ul> <p>마찬가지로, 다음 두 조건이 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. </p> <ul><li><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=1}^{\infty }|a_{n}\cos nx|<\infty }"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mi>n</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }|a_{n}\cos nx|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c656edf8023cf9de6f650ded0e28d3c456ab3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.516ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }|a_{n}\cos nx|<\infty }"></span></li> <li><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=1}^{\infty }|a_{n}|<\infty }"> <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>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }|a_{n}|<\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d363ec23ac87f20c95a403ec41b52f1ef63799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.906ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }|a_{n}|<\infty }"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="역사"><span id=".EC.97.AD.EC.82.AC"></span>역사</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=8" title="부분 편집: 역사"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>작자 <a href="/wiki/%ED%8E%98%ED%84%B0_%EA%B5%AC%EC%8A%A4%ED%83%80%ED%94%84_%EB%A5%B4%EC%A3%88_%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88" title="페터 구스타프 르죈 디리클레">페터 구스타프 르죈 디리클레</a>의 사후인 1862년 《<a href="/wiki/%EC%88%9C%EC%88%98_%EB%B0%8F_%EC%9D%91%EC%9A%A9%EC%88%98%ED%95%99_%EC%A0%80%EB%84%90" title="순수 및 응용수학 저널">순수 및 응용수학 저널</a>》(<span lang="fr">Journal de Mathématiques Pures et Appliquées</span>)에 게재되었다.<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="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=9" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%EC%95%84%EB%B2%A8_%ED%8C%90%EC%A0%95%EB%B2%95" title="아벨 판정법">아벨 판정법</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="각주"><span id=".EA.B0.81.EC.A3.BC"></span>각주</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=10" title="부분 편집: 각주"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-김락중-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-김락중_1-0">가</a></sup> <sup><a href="#cite_ref-김락중_1-1">나</a></sup></span> <span class="reference-text"><cite class="citation book">김락중; 박종안; 이춘호; 최규흥 (2007). 《해석학 입문》 3판. 경문사. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-8-96-105054-8" title="특수:책찾기/978-8-96-105054-8"><bdi>978-8-96-105054-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%ED%95%B4%EC%84%9D%ED%95%99+%EC%9E%85%EB%AC%B8&rft.edition=3&rft.pub=%EA%B2%BD%EB%AC%B8%EC%82%AC&rft.date=2007&rft.isbn=978-8-96-105054-8&rft.au=%EA%B9%80%EB%9D%BD%EC%A4%91&rft.au=%EB%B0%95%EC%A2%85%EC%95%88&rft.au=%EC%9D%B4%EC%B6%98%ED%98%B8&rft.au=%EC%B5%9C%EA%B7%9C%ED%9D%A5&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Knopp-2"><span class="mw-cite-backlink"><a href="#cite_ref-Knopp_2-0">↑</a></span> <span class="reference-text"><cite class="citation book">Knopp, Konrad (1951). 《Theory and application of infinite series》 (영어). 번역 Young, R. C. H.. Translated from the 2nd edition and revised in accordance with the fourth by R. C. H. Young. 2판. London–Glasgow: Blackie & Son. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0042.29203">0042.29203</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.place=London%E2%80%93Glasgow&rft.edition=2&rft.pub=Blackie+%26+Son&rft.date=1951&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0042.29203&rft.aulast=Knopp&rft.aufirst=Konrad&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><i>Démonstration d’un théorème d’Abel</i>. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), <a rel="nofollow" class="external text" href="http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0">pp. 253-255</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110721011902/http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1862_2_7_A43_0#">Archived</a> 2011년 7월 21일 - <a href="/wiki/%EC%9B%A8%EC%9D%B4%EB%B0%B1_%EB%A8%B8%EC%8B%A0" title="웨이백 머신">웨이백 머신</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95&action=edit&section=11" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Dirichlet_criterion_(convergence_of_series)">“Dirichlet criterion (convergence of series)”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Dirichlet+criterion+%28convergence+of+series%29&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FDirichlet_criterion_%28convergence_of_series%29&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DirichletsTest.html">“Dirichlet's test”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Dirichlet%27s+test&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDirichletsTest.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://planetmath.org/DirichletsConvergenceTest">“Dirichlet’s convergence test”</a>. 《PlanetMath》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=PlanetMath&rft.atitle=Dirichlet%E2%80%99s+convergence+test&rft_id=http%3A%2F%2Fplanetmath.org%2FDirichletsConvergenceTest&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Dirichlet%27s_Test_for_Uniform_Convergence">“Dirichlet's test for uniform convergence”</a>. 《ProofWiki》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ProofWiki&rft.atitle=Dirichlet%27s+test+for+uniform+convergence&rft_id=http%3A%2F%2Fproofwiki.org%2Fwiki%2FDirichlet%2527s_Test_for_Uniform_Convergence&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88+%ED%8C%90%EC%A0%95%EB%B2%95" class="Z3988"><span style="display:none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">원본 주소 "<a dir="ltr" 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디리클레 판정법 - 위키백과, 우리 모두의 백과사전