{"id":616,"date":"2018-06-23T06:55:16","date_gmt":"2018-06-23T06:55:16","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=616"},"modified":"2018-06-23T06:55:16","modified_gmt":"2018-06-23T06:55:16","slug":"jisuansum_n1inftyfrac1nanbncdezhi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=616","title":{"rendered":"\u8ba1\u7b97$\\sum_{n=1}^\\infty\\frac{1}{(n+a)(n+b)(n+c)}$\u7684\u503c"},"content":{"rendered":"

SE\u4e0a\u6709\u5173\u65e0\u7a77\u6c42\u548c(\u6b27\u62c9\u548c)
\n\\begin{equation}\\label{eq:n2}
\n\\sum_{n=1}^\\infty\\frac{1}{n^2}
\n\\end{equation}
\n\u7684\u8ba8\u8bba\u3002\u53c2\u8003Different methods to compute $\\sum_{k=1}^\\infty\\frac{1}{k^2}$ (Basel problem)<\/a>.<\/p>\n

\u6211\u7684\u95ee\u9898\u662f, \u5982\u4f55\u7528\u4ed6\u4eec\u7684\u529e\u6cd5\u6c42
\n\\begin{equation}\\label{eq:n3}
\n\\sum_{n=1}^\\infty\\frac{1}{(n+a)(n+b)(n+c)}
\n\\end{equation}
\n\u6ce8\u610f\u5230
\n$$
\n\\sum_{n=1}^\\infty\\frac{1}{n^3}
\n$$
\n\u662f$\\zeta(3)$\u5e76\u4e0d\u80fd\u51c6\u786e\u7b97\u51fa\u6765(\u4e0d\u80fd\u7528\u5df2\u77e5\u5e38\u6570\u8868\u793a)\u3002<\/p>\n

MMA\u6a21\u62df\u53ef\u77e5, \u5bf9\u4e0d\u5b8c\u5168\u76f8\u540c\u7684$a,b,c$, \\eqref{eq:n3}\u7684\u503c\u662f\u53ef\u4ee5\u5177\u4f53\u8ba1\u7b97\u7684\u3002<\/p>\n

\r\nmax = 3;\r\nTable[{a, b, c}, {a, 0, max}, {b, 0, max}, {c, b + 1, max}];\r\nlist = Flatten[\nDeleteDuplicates[\nf[a_, b_, c_] := {a, b, c, \r\n  Sum[1\/((n + a) (n + b) (n + c)), {n, 1, \\[Infinity]}]}\r\nApply[f, \n<\/code><\/pre>\n
\u5176\u7ed3\u679c\u5982\u4e0b
\n\\begin{array}{cccl}
\n a & b & c & \\sum\\limits_{n=1}^\\infty\\frac{1}{(n+a)(n+b)(n+c)}\\\\
\n\\hline
\n 0 & 0 & 1 & -1+\\frac{\\pi ^2}{6} \\\\
\n 0 & 0 & 2 & \\frac{1}{24} \\left(-9+2 \\pi ^2\\right) \\\\
\n 0 & 0 & 3 & \\frac{1}{54} \\left(-11+3 \\pi ^2\\right) \\\\
\n 0 & 1 & 2 & \\frac{1}{4} \\\\
\n 0 & 1 & 3 & \\frac{7}{36} \\\\
\n 0 & 2 & 3 & \\frac{5}{36} \\\\
\n 1 & 0 & 1 & 2-\\frac{\\pi ^2}{6} \\\\
\n 1 & 1 & 2 & \\frac{1}{6} \\left(-9+\\pi ^2\\right) \\\\
\n 1 & 1 & 3 & \\frac{1}{24} \\left(-17+2 \\pi ^2\\right) \\\\
\n 1 & 2 & 3 & \\frac{1}{12} \\\\
\n 2 & 0 & 2 & \\frac{1}{12} \\left(12-\\pi ^2\\right) \\\\
\n 2 & 1 & 2 & \\frac{1}{12} \\left(21-2 \\pi ^2\\right) \\\\
\n 2 & 2 & 3 & \\frac{1}{12} \\left(-19+2 \\pi ^2\\right) \\\\
\n 3 & 0 & 3 & \\frac{1}{324} \\left(213-18 \\pi ^2\\right) \\\\
\n 3 & 1 & 3 & \\frac{1}{36} \\left(32-3 \\pi ^2\\right) \\\\
\n 3 & 2 & 3 & \\frac{1}{36} \\left(61-6 \\pi ^2\\right) \\\\
\n\\end{array}<\/p>\n","protected":false},"excerpt":{"rendered":"
SE\u4e0a\u6709\u5173\u65e0\u7a77\u6c42\u548c(\u6b27\u62c9\u548c) \\begin{equation}\\label{eq:n2} \\sum_{n=1}… \u7ee7\u7eed\u9605\u8bfb\u8ba1\u7b97$\\sum_{n=1}^\\infty\\frac{1}{(n+a)(n+b)(n+c)}$\u7684\u503c<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[131,132],"class_list":["post-616","post","type-post","status-publish","format-standard","hentry","category-math","tag-wuqiongjishu","tag-qiuhe","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/616","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=616"}],"version-history":[{"count":11,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/616\/revisions"}],"predecessor-version":[{"id":627,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/616\/revisions\/627"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=616"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=616"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=616"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}