{"id":2910,"date":"2018-09-24T18:43:59","date_gmt":"2018-09-24T09:43:59","guid":{"rendered":"https:\/\/pandanote.info\/?p=2910"},"modified":"2022-08-07T12:03:17","modified_gmt":"2022-08-07T03:03:17","slug":"sqrtnn%e3%81%8cn%e2%86%92%e2%88%9e%e3%81%ae%e3%81%a8%e3%81%8d%e3%81%ab%e2%88%9e%e3%81%ab%e7%99%ba%e6%95%a3%e3%81%99%e3%82%8b%e3%81%93%e3%81%a8%e3%81%ae%e8%a8%bc%e6%98%8e%e3%80%82","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=2910","title":{"rendered":"\\sqrt[n]{n!}\u53ca\u3073\\sqrt[n]{n!!}\u304cn\u2192\u221e\u306e\u3068\u304d\u306b\u221e\u306b\u767a\u6563\u3059\u308b\u3053\u3068\u306e\u8a3c\u660e\u3002"},"content":{"rendered":"

\u306f\u3058\u3081\u306b<\/h2>\n

\u8a18\u4e8b\u306e\u30bf\u30a4\u30c8\u30eb\u306b\u3044\u304d\u306a\u308a$\\LaTeX$\u306e\u30b3\u30de\u30f3\u30c9\u306e\u3088\u3046\u306a\u3082\u306e\u3092\u5165\u308c\u3066\u3057\u307e\u3063\u3066\u3044\u3066\u3059\u307f\u307e\u305b\u3093\u304c\u3001\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u8a3c\u660e\u3092\u7701\u7565\u3057\u3066\u3057\u307e\u3063\u305f\u4ef6\u306b\u3064\u3044\u3066\u66f8\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n

\u3063\u3066\u3044\u3046\u304b\u3001\u8a08\u7b97\u7528\u7d19\u306b\u66f8\u3044\u3066\u307f\u308b\u3068\u2026
\n\"\"<\/a>
\n(\u203b\u5b57\u304c\u96d1\u306a\u306e\u3067\u3001\u30e2\u30b6\u30a4\u30af\u51e6\u7406\u3092\u3057\u3066\u3044\u307e\u3059\u3002)<\/p>\n

\u307e\u305f\u5f8c\u3067\u4f7f\u3044\u305d\u3046\u306a\u6c17\u304c\u3057\u3066\u304d\u305f\u306e\u3067\u3001\u30e1\u30e2\u66f8\u304d\u3057\u3066\u304a\u304f\u3053\u3068\u306b\u3057\u305f\u308f\u3051\u3067\u3059\u304c\u3001\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u8a3c\u660e\u304c\u5fc5\u8981\u3060\u3063\u305f\u5f0f\u3068\u306f\u7570\u306a\u308b\u5f0f\u3092\u6700\u521d\u306f\u8a3c\u660e\u3057\u3066\u3044\u305f\u306e\u3067\u3001\u672c\u6765\u8a3c\u660e\u3059\u3079\u304d\u5f0f\u306b\u3064\u3044\u3066\u306e\u8a3c\u660e\u3092\u554f\u984c(2)\u3068\u3057\u3066\u8ffd\u52a0\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u554f\u984c(1)<\/h2>\n

\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

$n$\u3092\u81ea\u7136\u6570\u3068\u3059\u308b\u3068\u304d\u3001$\\displaystyle\\lim_{n\\to\\infty}\\sqrt[n]{n!} = \\infty$\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088\u3002<\/p>\n

\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/h3>\n

\u307e\u305a\u3001$a_n=\\sqrt[n]{n!}$\u3068\u304a\u3044\u3066\u3001\u4e21\u8fba\u306e\u81ea\u7136\u5bfe\u6570\u3092\u53d6\u308a\u307e\u3059\u3002\u3059\u308b\u3068\u3001
\n\\begin{align}
\n\\log a_n &= \\log \\sqrt[n]{n!} \\nonumber \\\\
\n&= \\frac{1}{n} \\log n! \\nonumber \\\\
\n&= \\frac{1}{n} \\sum_{k=1}^{n}\\log k \\label{eq:logk}
\n\\end{align}
\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

$k=1$\u306e\u3068\u304d\u306b\u306f$\\log k=0$\u3068\u306a\u308b\u306e\u3067\u3001$k \\ge 2$\u306e\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u95a2\u6570$\\log x$\u306f$x \\gt 0$\u3067\u9023\u7d9a\u304b\u3064\u5358\u8abf\u5897\u52a0\u3067\u3059\u306e\u3067\u3001$k-1 \\le x \\le k$\u3067
\n\\begin{align}
\n\\log(k-1) &\\le \\log x \\le \\log k \\label{eq:k-1tok}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002\u6b21\u306b\u3001(\\ref{eq:k-1tok})\u5f0f\u306e\u5404\u8fba\u3092$[k-1,k]$\u306e\u533a\u9593\u3067$x$\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3068\u3001\u6700\u5de6\u8fba\u53ca\u3073\u6700\u53f3\u8fba\u306f$x$\u306b\u3064\u3044\u3066\u306f\u5b9a\u6570\u3067\u3059\u306e\u3067\u3001
\n\\begin{align}
\n\\log(k-1) &\\le \\int_{k-1}^k \\log x dx \\le \\log k \\label{eq:intk-1tok}
\n\\end{align}
\n\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

$k=1$\u306e\u3068\u304d\u306b\u306f$\\log k=0$\u3068\u306a\u308b\u3053\u3068\u3068\u3001(\\ref{eq:intk-1tok})\u5f0f\u306e2\u756a\u76ee\u306e\u4e0d\u7b49\u53f7\u306e\u4e21\u5074\u306e\u8fba\u306b\u7740\u76ee\u3057\u30012\u304b\u3089$n$\u307e\u3067\u306e\u548c\u3092\u3068\u308b\u3068\u3001
\n\\begin{align}
\n\\int_{1}^{n} \\log x dx &\\le \\sum_{k=2}^{n} \\log k \\nonumber \\\\
\n&= \\sum_{k=1}^{n} \\log k \\label{eq:intklogk}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001(\\ref{eq:intklogk})\u306e\u53f3\u8fba\u306f
\n\\begin{align}
\n\\int_{1}^{n} \\log x dx &= \\left[ x\\log x – x \\right]_1^n \\nonumber \\\\
\n&= n\\log n – n + 1 \\label{eq:logxdx}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u305d\u3053\u3067\u3001(\\ref{eq:logxdx})\u5f0f\u306e\u4e21\u8fba\u306b$\\displaystyle\\frac{1}{n}$\u3092\u304b\u3051\u3066\u3001\u3055\u3089\u306b(\\ref{eq:intklogk})\u5f0f\u53ca\u3073(\\ref{eq:logk})\u5f0f\u306e\u95a2\u4fc2\u3092\u5229\u7528\u3059\u308b\u3068\u3001
\n\\begin{align}
\n\\frac{1}{n}\\int_{1}^{n} \\log x dx &= \\log n – 1 + \\frac{1}{n} \\nonumber \\\\
\n&\\le \\frac{1}{n}\\sum_{k=1}^{n} \\log k = \\log a_n \\label{eq:evaluation}
\n\\end{align}<\/p>\n

$\\displaystyle\\lim_{n\\to\\infty}\\left(\\log n – 1 + \\frac{1}{n}\\right) = \\infty$\u3067\u3042\u308b\u3053\u3068\u3068\u3001(\\ref{eq:evaluation})\u5f0f\u306e\u95a2\u4fc2\u3088\u308a\u3001$\\displaystyle\\lim_{n\\to\\infty} \\log a_n = \\infty$\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001$\\displaystyle\\lim_{n\\to\\infty} a_n = \\infty$\u3068\u306a\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u554f\u984c(2)<\/h2>\n

\u4ee5\u4e0b\u306e\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u8d64\u6587\u5b57\u3067\u793a\u3057\u305f\u90e8\u5206\u304c\u554f\u984c(1)\u3068\u306f\u7570\u306a\u308b\u90e8\u5206\u3067\u3059\u3002<\/p>\n

$n$\u3092\u5076\u6570\u3068\u3059\u308b\u3068\u304d\u3001$\\displaystyle\\lim_{n\\to\\infty}\\sqrt[n]{\\color{red}{n!!}} = \\infty$\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088\u3002<\/p>\n

\u8a3c\u660e\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/h3>\n

$n=2m$\u3068\u304a\u304d\u3001\u3055\u3089\u306b$a_m=\\sqrt[2m]{2m!!}$\u3068\u304a\u304f\u3068\u3001
\n\\begin{align}
\na_m &= \\sqrt[2m]{2m!!} \\nonumber \\\\
\n&= \\sqrt[2m]{2^m m!} \\nonumber \\\\
\n&= \\sqrt{2\\sqrt[m]{m!}} \\label{eq:am}
\n\\end{align}
\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002$m$\u306f\u81ea\u7136\u6570\u3067\u3042\u308b\u304b\u3089\u3001\u554f\u984c(1)\u3088\u308a$\\displaystyle\\lim_{m\\to\\infty}\\sqrt[m]{m!} = \\infty$\u3067\u3042\u308b\u306e\u3067\u3001$\\displaystyle\\lim_{m\\to\\infty} a_m = \\infty$\u306b\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066$\\displaystyle\\lim_{n\\to\\infty}\\sqrt[n]{\\color{red}{n!!}} = \\infty$\u3068\u306a\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u307e\u3068\u3081<\/h2>\n

\u554f\u984c(1)\u53ca\u3073\u554f\u984c(2)\u306e\u5f0f\u3092\u898b\u308b\u3068\u76f4\u611f\u7684\u306b\u306f\u767a\u6563\u3057\u305d\u3046\u3060\u306a\u3068\u601d\u3046\u306e\u3067\u3059\u304c\u3001\u5b9f\u969b\u306b\u8a3c\u660e\u3059\u308b\u3068\u306a\u308b\u3068\u3001\u3053\u306e\u8a18\u4e8b\u306b\u66f8\u3044\u305f\u3088\u3046\u306a\u611f\u3058\u306e\u8a08\u7b97\u3092\u884c\u3046\u3053\u3068\u306b\u306a\u308b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n

\u5b9f\u306f$n$\u4e57\u6839\u304c$\\LaTeX$\u3067\u306f\sqrt[n]{}\u306e\u3088\u3046\u306a\u66f8\u304d\u65b9\u304c\u3067\u304d\u308b\u3053\u3068\u306b\u3053\u306e\u8a18\u4e8b\u3092\u66f8\u3044\u3066\u3044\u3066\u6c17\u304c\u4ed8\u304d\u307e\u3057\u305f\u3002(\u00b4\u30fb\u03c9\u30fb\uff40)<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u306f\u3058\u3081\u306b \u8a18\u4e8b\u306e\u30bf\u30a4\u30c8\u30eb\u306b\u3044\u304d\u306a\u308a$\\LaTeX$\u306e\u30b3\u30de\u30f3\u30c9\u306e\u3088\u3046\u306a\u3082\u306e\u3092\u5165\u308c\u3066\u3057\u307e\u3063\u3066\u3044\u3066\u3059\u307f\u307e\u305b\u3093\u304c\u3001\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b\u3067\u8a3c\u660e\u3092\u7701\u7565\u3057\u3066\u3057\u307e\u3063\u305f\u4ef6\u306b\u3064\u3044\u3066\u66f8\u304d\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002 \u3063\u3066\u3044\u3046\u304b\u3001\u8a08\u7b97\u7528\u7d19\u306b\u66f8\u3044\u3066\u307f\u308b\u3068\u2026 (\u203b\u5b57\u304c\u96d1\u306a\u306e\u3067\u3001\u30e2\u30b6\u30a4\u30af\u51e6\u7406\u3092\u3057\u3066\u3044\u307e\u3059\u3002) \u307e\u305f\u5f8c\u3067\u4f7f\u3044\u305d\u3046\u306a\u6c17\u304c\u3057\u3066\u304d\u305f\u306e\u3067\u3001\u30e1\u30e2\u66f8\u304d\u3057\u3066\u304a\u304f\u3053\u3068\u306b\u3057\u305f\u308f\u3051\u3067\u3059\u304c\u3001\u3061\u3087\u3063\u3068\u524d\u306e\u8a18\u4e8b\u3067\u8a3c\u660e\u304c\u5fc5\u8981\u3060\u3063\u305f\u5f0f\u3068\u306f\u7570\u306a\u308b\u5f0f\u3092\u6700\u521d\u306f\u8a3c\u660e\u3057\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":2931,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5,13],"tags":[],"class_list":["post-2910","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-latex","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/2910","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2910"}],"version-history":[{"count":30,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/2910\/revisions"}],"predecessor-version":[{"id":9358,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/2910\/revisions\/9358"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/2931"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2910"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2910"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}