{"id":1502,"date":"2025-08-12T22:26:21","date_gmt":"2025-08-12T13:26:21","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=1502"},"modified":"2025-08-12T22:26:22","modified_gmt":"2025-08-12T13:26:22","slug":"post-1502","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=1502","title":{"rendered":"\u30d9\u30eb\u30cc\u30fc\u30a4\u6570"},"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=1502\/#%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=1502\/#%E3%83%9E%E3%82%AF%E3%83%AD%E3%83%BC%E3%83%AA%E3%83%B3%E5%B1%95%E9%96%8B%E3%81%8B%E3%82%89%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E6%95%B0%E3%82%92%E8%A8%88%E7%AE%97%E3%81%99%E3%82%8B\" >\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u304b\u3089\u30d9\u30eb\u30cc\u30fc\u30a4\u6570\u3092\u8a08\u7b97\u3059\u308b<\/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=1502\/#%E6%BC%B8%E5%8C%96%E5%BC%8F\" >\u6f38\u5316\u5f0f<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/daba-no-heya.com\/?p=1502\/#%E8%A8%BC%E6%98%8E\" >\u8a3c\u660e<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/daba-no-heya.com\/?p=1502\/#n%E3%82%923%E4%BB%A5%E4%B8%8A%E3%81%AE%E5%A5%87%E6%95%B0%E3%81%A8%E3%81%99%E3%82%8B%E3%81%A8B_n0\" >$n$\u30923\u4ee5\u4e0a\u306e\u5947\u6570\u3068\u3059\u308b\u3068$B_n=0$<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/daba-no-heya.com\/?p=1502\/#%E8%A8%BC%E6%98%8E-2\" >\u8a3c\u660e<\/a><\/li><\/ul><\/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<p>\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$B_n$\u3092\u30d9\u30eb\u30cc\u30fc\u30a4\u6570(Bernoulli number)\u3068\u547c\u3073\u307e\u3059\u3002<br>$\\frac{x}{e^x-1}$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u305f\u3068\u304d\u306e\u4fc2\u6570\u3067\u3059\u3002<\/p>\n\n\n\n<p>$$<br>\\frac{x}{e^x-1}=\\sum^{\\infty}_{n=0}\\frac{B_n}{n!}x^n<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%9E%E3%82%AF%E3%83%AD%E3%83%BC%E3%83%AA%E3%83%B3%E5%B1%95%E9%96%8B%E3%81%8B%E3%82%89%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E6%95%B0%E3%82%92%E8%A8%88%E7%AE%97%E3%81%99%E3%82%8B\"><\/span>\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u304b\u3089\u30d9\u30eb\u30cc\u30fc\u30a4\u6570\u3092\u8a08\u7b97\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$\\frac{x}{e^x-1}$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3059\u308b\u3053\u3068\u3067\u3001$B_n$\u306e\u5024\u3092\u76f4\u63a5\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p>$e^x-1$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3059\u308b\u3068<\/p>\n\n\n\n<p>$$<br>e^x-1=x+\\frac{1}{2!}x^2+\\frac{1}{3!}x^3+\\cdots<br>$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\frac{x}{e^x-1}&amp;=\\frac{x}{x+\\frac{1}{2!}x^2+\\frac{1}{3!}x^3+\\cdots} \\\\<br>&amp;=\\frac{1}{1+\\frac{1}{2!}x+\\frac{1}{3!}x^2+\\cdots} \\\\<br>&amp;=\\frac{1}{1+\\left(\\frac{1}{2!}x+\\frac{1}{3!}x^2+\\cdots\\right)}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>$\\frac{1}{1+x}$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3059\u308b\u3068<\/p>\n\n\n\n<p>$$<br>\\frac{1}{1+x}=1-x+x^2-x^3+\\cdots<br>$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\frac{x}{e^x-1}&amp;=1-\\left(\\frac{1}{2!}x+\\frac{1}{3!}x^2+\\cdots\\right)+\\left(\\frac{1}{2!}x+\\frac{1}{3!}x^2+\\cdots\\right)^2-\\cdots \\\\<br>&amp;=1-\\frac{1}{2!}x+\\left(-\\frac{1}{3!}+\\left(\\frac{1}{2!}\\right)^2\\right)x^2+\\cdots \\\\<br>&amp;=1-\\frac{1}{2}x+\\frac{1}{12}x^2+\\cdots<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>\u5143\u306e\u5b9a\u7fa9\u5f0f\u3068\u6bd4\u8f03\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>&amp;B_0=1 \\\\<br>&amp;B_1=-\\frac{1}{2} \\\\<br>&amp;B_2=\\frac{1}{6}<br>\\end{align}<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%BC%B8%E5%8C%96%E5%BC%8F\"><\/span>\u6f38\u5316\u5f0f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u4ee5\u4e0b\u306e\u6f38\u5316\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br>\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u304b\u3089\u30d9\u30eb\u30cc\u30fc\u30a4\u6570\u3092\u8a08\u7b97\u3059\u308b\u3088\u308a\u3082\u3053\u3061\u3089\u306e\u6f38\u5316\u5f0f\u3092\u7528\u3044\u308b\u65b9\u304c\u7c21\u5358\u3060\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p>$$<br>B_n=-\\frac{1}{n+1}\\sum_{k=0}^{n-1}{}_{n+1}C_kB_k<br>$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%A8%BC%E6%98%8E\"><\/span>\u8a3c\u660e<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>$$<br>\\begin{align}<br>1&amp;=\\frac{x}{e^x-1}\\cdot\\frac{e^x-1}{x} \\\\<br>&amp;=\\left(\\sum_{n=0}^{\\infty}\\frac{B_n}{n!}x^n\\right)\\left(\\sum_{n=1}^{\\infty}\\frac{1}{n!}x^{n-1}\\right) \\\\<br>&amp;=\\left(B_0+\\frac{B_1}{1!}x+\\frac{B_2}{2!}x^2+\\frac{B_3}{3!}x^3+\\cdots\\right)\\left(\\frac{1}{1!}+\\frac{1}{2!}x+\\frac{1}{3!}x^2+\\frac{1}{4!}x^3+\\cdots\\right) \\\\<br>&amp;=\\frac{B_0}{1!}+\\left(\\frac{B_0}{2!}+\\frac{B_1}{1!1!}\\right)x+\\left(\\frac{B_0}{3!}+\\frac{B_1}{1!2!}+\\frac{B_2}{2!1!}\\right)x^2 \\\\<br>&amp;\\qquad+\\left(\\frac{B_0}{4!}+\\frac{B_1}{1!3!}+\\frac{B_2}{2!2!}+\\frac{B_3}{3!1!}\\right)x^3+\\cdots \\\\<br>&amp;=\\sum_{k=0}^{0}\\frac{B_k}{\\left(1-k\\right)!k!}+\\left(\\sum_{k=0}^{1}\\frac{B_k}{\\left(2-k\\right)!k!}\\right)x \\\\<br>&amp;\\qquad+\\left(\\sum_{k=0}^{2}\\frac{B_k}{\\left(3-k\\right)!k!}\\right)x^2+\\left(\\sum_{k=0}^{3}\\frac{B_k}{\\left(4-k\\right)!k!}\\right)x^3+\\cdots \\\\<br>&amp;=\\sum_{n=0}^{\\infty}\\left(\\sum_{k=0}^{n}\\frac{B_k}{\\left(n+1-k\\right)!k!}\\right)x^n<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>$$<br>{}_{n+1}C_k=\\frac{\\left(n+1\\right)!}{k!\\left(n+1-k\\right)!}<br>$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>\\sum_{n=0}^{\\infty}\\left(\\sum_{k=0}^{n}\\frac{B_k}{\\left(n+1-k\\right)!k!}\\right)x^n=\\sum_{n=0}^{\\infty}\\left(\\sum_{k=0}^{n}\\frac{1}{\\left(n+1\\right)!}{}_{n+1}C_kB_k\\right)x^n<br>$$<\/p>\n\n\n\n<p>$n\\ge1$\u306b\u3064\u3044\u3066$x^n$\u306e\u9805\u306f\u73fe\u308c\u306a\u3044\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>\\sum_{k=0}^n\\frac{1}{\\left(n+1\\right)!}{}_{n+1}C_kB_k=0<br>$$<\/p>\n\n\n\n<p>\u3088\u308a\u3001<\/p>\n\n\n\n<p>$$<br>\\sum_{k=0}^{n}{}_{n+1}C_kB_k=0<br>$$<\/p>\n\n\n\n<p>\u5f0f\u3092\u6574\u7406\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\sum_{k=0}^{n}{}_{n+1}C_kB_k&amp;=\\sum_{k=0}^{n-1}{}_{n+1}C_kB_k+{}_{n+1}C_nB_n=0 \\\\<br>{}_{n+1}C_nB_n&amp;=-\\sum_{k=0}^{n-1}{}_{n+1}C_kB_k<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p>$$<br>B_n=-\\frac{1}{n+1}\\sum_{k=0}^{n-1}{}_{n+1}C_kB_k<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"n%E3%82%923%E4%BB%A5%E4%B8%8A%E3%81%AE%E5%A5%87%E6%95%B0%E3%81%A8%E3%81%99%E3%82%8B%E3%81%A8B_n0\"><\/span>$n$\u30923\u4ee5\u4e0a\u306e\u5947\u6570\u3068\u3059\u308b\u3068$B_n=0$<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>$n\\ge3$\u306e\u5947\u6570\u3068\u3059\u308b\u3068\u3001$B_n=0$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%A8%BC%E6%98%8E-2\"><\/span>\u8a3c\u660e<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>$\\frac{x}{e^x-1}$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u305f\u3068\u304d\u306b\u73fe\u308c\u308b\u5947\u6570\u6b21\u306e\u9805\u306f1\u6b21\u306e$-\\frac{1}{2}x$\u3060\u3051\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059\u3002<br>\u3064\u307e\u308a\u3001$\\frac{x}{e^x-1}+\\frac{1}{2}x$\u304c\u5076\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308c\u3070\u3044\u3044\u3002<\/p>\n\n\n\n<p>$$<br>\\begin{align}<br>\\frac{-x}{e^{-x}-1}-\\frac{1}{2}x&amp;=\\frac{x}{1-e^{-x}}-\\frac{1}{2}x \\\\<br>&amp;=\\frac{2-\\left(1-e^{-x}\\right)}{2\\left(1-e^{-x}\\right)}x \\\\<br>&amp;=\\frac{1+e^{-x}}{2\\left(1-e^{-x}\\right)}x \\\\<br>&amp;=\\frac{e^x+1}{2\\left(e^x-1\\right)}x \\\\<br>&amp;=\\frac{x}{e^x-1}+\\frac{1}{2}x<br>\\end{align}<br>$$<\/p>\n\n\n\n<p>\u3057\u305f\u304c\u3063\u3066\u3001$\\frac{x}{e^x-1}+\\frac{1}{2}x$\u306f\u5076\u95a2\u6570\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p class=\".scroll\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6982\u8981 \u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b$B_n$\u3092\u30d9\u30eb\u30cc\u30fc\u30a4\u6570(Bernoulli number)\u3068\u547c\u3073\u307e\u3059\u3002$\\frac{x}{e^x-1}$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3057\u305f\u3068\u304d\u306e\u4fc2\u6570\u3067\u3059\u3002 $$\\frac{x}{e^x-1}=\\sum^ [&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-1502","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\/1502","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=1502"}],"version-history":[{"count":19,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1502\/revisions"}],"predecessor-version":[{"id":1529,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1502\/revisions\/1529"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1502"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1502"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1502"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}