{"id":1479,"date":"2025-06-15T21:21:26","date_gmt":"2025-06-15T12:21:26","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=1479"},"modified":"2025-08-13T20:46:28","modified_gmt":"2025-08-13T11:46:28","slug":"post-1479","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=1479","title":{"rendered":"\u30d0\u30fc\u30bc\u30eb\u554f\u984c"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/daba-no-heya.com\/?p=1479\/#%E6%A6%82%E8%A6%81\" >\u6982\u8981<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/daba-no-heya.com\/?p=1479\/#sum_n1infty_frac1n2%E3%81%8C%E5%8F%8E%E6%9D%9F%E3%81%99%E3%82%8B%E3%81%8B\" >$\\sum_{n=1}^\\infty \\frac{1}{n^2}$\u304c\u53ce\u675f\u3059\u308b\u304b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/daba-no-heya.com\/?p=1479\/#%E5%8F%8E%E6%9D%9F%E3%81%99%E3%82%8B%E5%A0%B4%E5%90%88%E3%81%AE%E5%80%A4%E3%81%AF%E4%BD%95%E3%81%8B\" >\u53ce\u675f\u3059\u308b\u5834\u5408\u306e\u5024\u306f\u4f55\u304b<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%A6%82%E8%A6%81\"><\/span>\u6982\u8981<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$\\sum_{n=1}^\\infty \\frac{1}{n^2}$\u304c\u53ce\u675f\u3059\u308b\u304b<\/li>\n\n\n\n<li>\u53ce\u675f\u3059\u308b\u5834\u5408\u306e\u5024\u306f\u4f55\u304b<\/li>\n<\/ul>\n\n\n\n<p>\u3068\u3044\u3046\u554f\u984c\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002<br>\u3053\u306e\u554f\u984c\u3092\u30d0\u30fc\u30bc\u30eb\u554f\u984c(Basel problem)\u3068\u547c\u3073\u307e\u3059\u3002<br>(\u53c2\u8003: <a href=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%90%E3%83%BC%E3%82%BC%E3%83%AB%E5%95%8F%E9%A1%8C\" data-type=\"link\" data-id=\"https:\/\/ja.wikipedia.org\/wiki\/%E3%83%90%E3%83%BC%E3%82%BC%E3%83%AB%E5%95%8F%E9%A1%8C\">Wikipedia<\/a>)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"sum_n1infty_frac1n2%E3%81%8C%E5%8F%8E%E6%9D%9F%E3%81%99%E3%82%8B%E3%81%8B\"><\/span>$\\sum_{n=1}^\\infty \\frac{1}{n^2}$\u304c\u53ce\u675f\u3059\u308b\u304b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$$<br>\\frac{1}{n^2}&lt;\\frac{1}{n(n-1)}\\quad(n\\geq 2)<br>$$<\/p>\n\n\n\n<p>\u306e\u95a2\u4fc2\u5f0f\u3092\u7528\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\sum_{n=1}^{\\infty}\\frac{1}{n^2}&amp;=1+\\sum_{n=2}^{\\infty}\\frac{1}{n^2} \\\\<br>&amp;&lt;1+\\sum_{n=2}^{\\infty}\\frac{1}{n(n-1)} \\\\<br>&amp;=1+\\sum_{n=2}^{\\infty}\\left(\\frac{1}{n-1}-\\frac{1}{n} \\right) \\\\<br>&amp;=1+\\left(\\frac{1}{1}-\\frac{1}{2}\\right)+\\left(\\frac{1}{2}-\\frac{1}{3}\\right)+\\left(\\frac{1}{3}-\\frac{1}{4}\\right)+\\cdots \\\\<br>&amp;=2<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001\u3053\u306e\u7d1a\u6570\u306f\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%8F%8E%E6%9D%9F%E3%81%99%E3%82%8B%E5%A0%B4%E5%90%88%E3%81%AE%E5%80%A4%E3%81%AF%E4%BD%95%E3%81%8B\"><\/span>\u53ce\u675f\u3059\u308b\u5834\u5408\u306e\u5024\u306f\u4f55\u304b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$\\sin x$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\sin x=\\frac{x^1}{1!}-\\frac{x^3}{3!}+\\frac{x^5}{5!}-\\frac{x^7}{7!}+\\cdots<br>$$<\/p>\n\n\n\n<p>\u3053\u308c\u3088\u308a\u3001$\\frac{\\sin x}{x}$\u306f\u3001<\/p>\n\n\n\n<p>$$<br>\\frac{\\sin x}{x}=\\frac{1}{1!}-\\frac{x^2}{3!}+\\frac{x^4}{5!}-\\frac{x^6}{7!}+\\cdots \\tag{1}<br>$$<\/p>\n\n\n\n<p>\u5de6\u8fba\u306f$x=\\pm n\\pi$ ($n$\u306f\u81ea\u7136\u6570)\u306e\u3068\u304d0\u306b\u306a\u308b\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\frac{\\sin x}{x}&amp;=\\left(1-\\frac{x}{1\\pi}\\right)\\left(1+\\frac{x}{1\\pi}\\right)\\left(1-\\frac{x}{2\\pi}\\right)\\left(1+\\frac{x}{2\\pi}\\right)\\left(1-\\frac{x}{3\\pi}\\right)\\left(1+\\frac{x}{3\\pi}\\right)\\cdots \\\\<br>&amp;=\\left(1-\\frac{x^2}{1^2\\pi^2}\\right)\\left(1-\\frac{x^2}{2^2\\pi^2}\\right)\\left(1-\\frac{x^2}{3^2\\pi^2}\\right)\\cdots \\tag{2}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p>\u5f0f1\u3068\u5f0f2\u3067$x^2$\u306e\u4fc2\u6570\u3092\u6bd4\u8f03\u3057\u307e\u3059\u3002<br>\u5f0f1\u306e$x^2$\u306e\u4fc2\u6570\u306f\u3001<\/p>\n\n\n\n<p>$$<br>-\\frac{1}{3!}=-\\frac{1}{6}<br>$$<\/p>\n\n\n\n<p>\u5f0f2\u306e$x^2$\u306e\u4fc2\u6570\u306f\u3001<\/p>\n\n\n\n<p>$$<br>-\\left(\\frac{1}{1^2\\pi^2}+\\frac{1}{2^2\\pi^2}+\\frac{1}{3^2\\pi^2}+\\cdots\\right)=-\\frac{1}{\\pi^2}\\sum_{n=1}^{\\infty}\\frac{1}{n^2}<br>$$<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p>$$<br>-\\frac{1}{\\pi^2}\\sum_{n=1}^{\\infty}\\frac{1}{n^2}=-\\frac{1}{6}<br>$$<\/p>\n\n\n\n<p>\u3088\u308a\u3001<\/p>\n\n\n\n<p>$$<br>\\sum_{n=1}^{\\infty}\\frac{1}{n^2}=\\frac{\\pi^2}{6}<br>$$<\/p>\n\n\n\n<p>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u8981 \u3068\u3044\u3046\u554f\u984c\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u554f\u984c\u3092\u30d0\u30fc\u30bc\u30eb\u554f\u984c(Basel problem)\u3068\u547c\u3073\u307e\u3059\u3002(\u53c2\u8003: Wikipedia) $\\sum_{n=1}^\\infty \\frac{1}{n^2}$\u304c\u53ce\u675f\u3059\u308b\u304b $$\\fr [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-1479","post","type-post","status-publish","format-standard","hentry","category-30"],"_links":{"self":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1479","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1479"}],"version-history":[{"count":10,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1479\/revisions"}],"predecessor-version":[{"id":1531,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1479\/revisions\/1531"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1479"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1479"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1479"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}