{"id":1482,"date":"2015-12-07T18:00:35","date_gmt":"2015-12-07T09:00:35","guid":{"rendered":"http:\/\/math.co.jp\/blog\/?p=1482"},"modified":"2015-12-07T17:45:12","modified_gmt":"2015-12-07T08:45:12","slug":"%e3%80%8c%e7%a4%ba%e3%81%9b%e3%80%8d%e3%81%a8%e3%80%8c%e6%b1%82%e3%82%81%e3%82%88%e3%80%8d%e3%81%af%e7%95%b0%e3%81%aa%e3%82%8b%e5%95%8f%e9%a1%8c","status":"publish","type":"post","link":"https:\/\/math.co.jp\/blog\/?p=1482","title":{"rendered":"\u300c\u793a\u305b\u300d\u3068\u300c\u6c42\u3081\u3088\u300d\u306f\u7570\u306a\u308b\u554f\u984c"},"content":{"rendered":"<p>\u3000\u4eca\u56de\u306f, \u5f0f\u5909\u5f62\u306e\u624b\u6cd5\u306b\u3064\u3044\u3066\u6271\u3044\u307e\u3059\u3002\u307e\u305a\u306f, \u6b21\u306e\u554f\u984c\u306b\u53d6\u308a\u7d44\u3093\u3067\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n<fieldset>\n<legend>\u3010\u554f\u984c C &#8211; 1 \u3011<\/legend>\n<p>\u3000\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3057, \u6570\u5217 \\(\\{a_{n}\\}\\) \u3092\u6b21\u5f0f\u3067\u5b9a\u7fa9\u3059\u308b. <\/p>\n<p>$$a_{n}=\\displaystyle\\int_{1}^{e}(\\log x)^{n}\\,dx\u3000(n=1,2,\\cdots)$$\u3000<br \/>\n(1)\u3000\\(n\\geqq 3\\) \u306e\u3068\u304d, \u6b21\u306e\u6f38\u5316\u5f0f\u3092\u793a\u305b.<br \/>\n$$a_{n}=(n-1)(a_{n-2}-a_{n-1})$$<br \/>\n<\/fieldset>\n<p>\u3053\u308c\u306f, 2005 \u5e74\u306e\u6771\u5de5\u5927\u306e\u554f\u984c\u3067\u3059\u3002\u5b9f\u969b\u306f, (2), (3) \u304c\u3042\u3063\u3066\u305d\u308c\u306f\u6b21\u306e\u3088\u3046\u306a\u554f\u984c\u3067\u3057\u305f\u3002<br \/>\n\u3000<br \/>\n(2)\u3000\\(n\\geqq 1\\) \u306b\u5bfe\u3057 \\(a_{n}\\gt a_{n+1}\\gt 0\\) \u306a\u308b\u3053\u3068\u3092\u793a\u305b.<br \/>\n(3)\u3000\\(n\\geqq 2\\) \u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<br \/>\n$$a_{2n}\\lt\\displaystyle\\frac{3\\cdot 5\\cdots (2n-1)}{4\\cdot 6\\cdots (2n)}(e-2)$$<br \/>\n\u4eca\u56de\u306f, (2), (3) \u306b\u89e6\u308c\u305a (1) \u306e\u5909\u5f62\u306b\u3064\u3044\u3066\u89e3\u8aac\u3059\u308b\u3053\u3068\u3068\u3057\u307e\u3059\u3002<br \/>\n\u3000<br \/>\n\u3000\u3055\u3066, \u300c\u793a\u305b\u300d\u3068\u8a00\u3063\u3066\u3044\u308b\u5f0f\u306f, \\(a_{n}\\) \u3092 \\(a_{n-1}\\), \\(a_{n-2}\\) \u3092\u7528\u3044\u3066\u8868\u3059\u5f0f\u3067\u3059\u3002\u305d\u3053\u3067, \\(a_{n}\\) \u3068 \\(a_{n-1}\\) \u306e\u95a2\u4fc2\u304c\u5fc5\u8981\u306b\u306a\u308b\u3053\u3068\u304b\u3089, \\(a_{n}\\) \u304a\u3088\u3073 \\(a_{n-1}\\) \u3092\u5177\u4f53\u7684\u306b\u66f8\u3044\u3066\u307f\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<br \/>\n\u3000<br \/>\n\u3000\u3000\\(a_{n}=\\displaystyle\\int_{1}^{e}(\\log x)^{n}\\,dx\\)<br \/>\n\u3000\u3000\\(a_{n-1}=\\displaystyle\\int_{1}^{e}(\\log x)^{n-1}\\,dx\\)<br \/>\n\u3000<br \/>\n\u7a4d\u5206\u8a18\u53f7\u5185\u306e \\((\\log x)^{n}\\) \u3092 \\((\\log x)^{n-1}\\) \u306b\u5909\u3048\u308b\u306b\u306f\u300c<strong>\u5fae\u5206\u3059\u308c\u3070\u3088\u3044<\/strong>\u300d\u3068\u8003\u3048\u3066\u6b21\u306e\u3088\u3046\u306b\u90e8\u5206\u7a4d\u5206\u3092\u884c\u306a\u3044\u307e\u3059\u3002 <\/p>\n<p>\\(\\begin{align}<br \/>\n\u3000\u3000a_{n}&#038;=\\displaystyle\\int_{1}^{e}1\\cdot (\\log x)^{n}\\,dx\\\\<br \/>\n\u3000\u3000\u3000&#038;=\\Bigl[\\,x(\\log x)^{n}\\Bigr]_{1}^{e}-\\displaystyle\\int_{1}^{e}x\\cdot n(\\log x)^{n-1}\\cdot \\displaystyle\\frac{1}{x}\\,dx\\\\<br \/>\n\u3000\u3000\u3000&#038;=e-n\\displaystyle\\int_{1}^{e}(\\log x)^{n-1}\\,dx\\\\<br \/>\n\u3000\u3000\u3000&#038;=e-na_{n-1}\\\\<br \/>\n\\end{align}\\)<\/p>\n<p>\u3059\u306a\u308f\u3061, <\/p>\n<p>\u3000\u3000\u3000\\(a_{n}=e-na_{n-1}\\)\u3000\\(\\cdots\\cdots\\,\\)\u2460<\/p>\n<p>\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u540c\u69d8\u306b\u756a\u53f7\u3092 1 \u3064\u4e0b\u3052\u308b\u3053\u3068\u3067, <\/p>\n<p>\u3000\u3000\u3000\\(a_{n-1}=e-(n-1)a_{n-2}\\)\u3000\\(\\cdots\\cdots\\,\\)\u2461<\/p>\n<p>\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<br \/>\n\u3000\u3055\u3066, \u3053\u3053\u304b\u3089\u304c\u4eca\u56de\u306e\u30c6\u30fc\u30de\u3067\u3059\u3002\u3053\u308c\u307e\u3067\u5f97\u3089\u308c\u3066\u3044\u308b\u2460\u3068\u2461\u304b\u3089\u3069\u306e\u3088\u3046\u306b\u3057\u3066, \\(a_{n}=(n-1)(a_{n-2}-a_{n-1})\\) \u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3067\u3057\u3087\u3046\u304b?<br \/>\n\u3000\u4e00\u822c\u306b, \u300c\u6c42\u3081\u3088\u300d\u3068\u3044\u3046\u30bf\u30a4\u30d7\u306e\u554f\u984c\u306f\u7d50\u679c\u306e\u6570\u5024\u304c\u308f\u304b\u3063\u3066\u3044\u307e\u305b\u3093\u3002\u3053\u308c\u306b\u5bfe\u3057, \u300c\u793a\u305b\u300d\u3068\u3044\u3046\u30bf\u30a4\u30d7\u306e\u554f\u984c\u306f\u7d50\u679c\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u306e\u3067, \u7d50\u679c\u306e\u5f62\u3092\u89b3\u5bdf\u3059\u308b\u3053\u3068\u3067\u73fe\u72b6\u304b\u3089\u4f55\u3092\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u304b\u304c\u308f\u304b\u308b\u5834\u5408\u304c\u3042\u308a\u307e\u3059\u3002\u5f0f\u5909\u5f62\u306b\u304a\u3044\u3066\u306f, \u300c<strong>\u793a\u305b\u3068\u3055\u308c\u308b\u5f0f\u306b\u542b\u307e\u308c\u308b\u6587\u5b57<\/strong>\u300d\u306b\u7740\u76ee\u3059\u308b\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002<br \/>\n\u3000\u307e\u305a, \u2460\u304a\u3088\u3073\u2461\u306b\u542b\u307e\u308c\u3066\u3044\u308b\u6587\u5b57\u3092\u898b\u307e\u3059\u3002\u305d\u308c\u306f,<br \/>\n\u3000\u3000\\(a_{n}\\), \\(a_{n-1}\\), \\(a_{n-2}\\), \\(n\\), \\(e\\)<br \/>\n\u3067\u3059\u3002\u3053\u308c\u306b\u5bfe\u3057\u300c\u793a\u305b\u300d\u3068\u3055\u308c\u308b\u5f0f\u306b\u542b\u307e\u308c\u308b\u6587\u5b57\u306f,<br \/>\n\u3000\u3000\\(a_{n}\\), \\(a_{n-1}\\), \\(a_{n-2}\\), \\(n\\)<br \/>\n\u3067\u3059\u3002\u305d\u308c\u3067\u306f\u524d\u8005\u306b\u3042\u3063\u3066\u5f8c\u8005\u306b\u306a\u3044\u6587\u5b57\u306f\u3069\u308c\u304b\u3068\u3044\u3046\u3068 \\(e\\) \u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066, \u2460 \u3068 \u2461 \u304b\u3089\u5f0f\u5909\u5f62\u306e\u3069\u3053\u304b\u3067 \\(e\\) \u3092\u6d88\u53bb\u3057\u306a\u3051\u308c\u3070\u793a\u3057\u305f\u3044\u5f0f\u306b\u306f\u7d76\u5bfe\u306b\u5230\u9054\u3057\u307e\u305b\u3093\u3002\u305d\u3053\u3067, \u6c42\u3081\u305f\u3044\u5f0f\u3092\u5f97\u308b\u305f\u3081\u306b\u300c\\(e\\) \u306e\u6d88\u53bb\u300d\u3092\u8003\u3048\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\\(e\\) \u3092\u6d88\u53bb\u3059\u308b\u305f\u3081\u306b\u4eca\u56de\u306f \u2460 \u304b\u3089 \u2461 \u3092\u5f15\u3044\u3066<br \/>\n\u3000\u3000\u3000\\(a_{n}-a_{n-1}=-na_{n-1}+(n-1)a_{n-2}\\)<br \/>\n\u3068\u3057, \u3053\u308c\u3092\u5909\u5f62\u3057\u3066<br \/>\n\u3000\u3000\u3000\\(a_{n}=(n-1)(a_{n-2}-a_{n-1})\\)<br \/>\n\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br \/>\n\u3000<br \/>\n\u3000\u4ee5\u4e0a\u306e\u3088\u3046\u306b\u300c\u793a\u305b\u300d\u3068\u3044\u3046\u554f\u984c\u306b\u5bfe\u3059\u308b\u65b9\u91dd\u3068\u3057\u3066, \u6b21\u306e\u3088\u3046\u306a\u3082\u306e\u304c\u6709\u52b9\u306a\u5834\u5408\u3082\u3042\u308b\u3068\u3044\u3046\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<p><strong>\u3000\u300c\u793a\u305b\u300d\u3068\u3044\u3046\u554f\u984c\u3067\u7d50\u8ad6\u306e\u5f0f\u304c\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u5834\u5408\u306f, \u7d50\u8ad6\u306e\u5f0f\u306b\u542b\u307e\u308c\u3066\u3044\u308b\u6587\u5b57\u306b\u6ce8\u76ee\u3057, \u6b8b\u3063\u3066\u3044\u3066\u306f\u3044\u3051\u306a\u3044\u6587\u5b57\u3092\u898b\u3064\u3051\u3088\u3002\u305d\u306e\u6587\u5b57\u3092\u6d88\u53bb\u3059\u308b\u65b9\u91dd\u3067\u8a08\u7b97\u51e6\u7406\u3092\u884c\u306a\u3046\u3068\u3046\u307e\u304f\u3044\u304f\u3053\u3068\u304c\u3042\u308b\u3002<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u3000\u4eca\u56de\u306f, \u5f0f\u5909\u5f62\u306e\u624b\u6cd5\u306b\u3064\u3044\u3066\u6271\u3044\u307e\u3059\u3002\u307e\u305a\u306f, \u6b21\u306e\u554f\u984c\u306b\u53d6\u308a\u7d44\u3093\u3067\u307f\u3066\u304f\u3060\u3055\u3044\u3002 \u3010\u554f\u984c C &#8211; 1 \u3011 \u3000\\(e\\) \u3092\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3057, \u6570\u5217 \\(\\{a_{n}\\}\\) \u3092\u6b21\u5f0f\u3067\u5b9a\u7fa9\u3059\u308b. $$a [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[22],"class_list":["post-1482","post","type-post","status-publish","format-standard","hentry","category-11","tag-etude-strategy"],"_links":{"self":[{"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1482","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1482"}],"version-history":[{"count":8,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1482\/revisions"}],"predecessor-version":[{"id":1491,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=\/wp\/v2\/posts\/1482\/revisions\/1491"}],"wp:attachment":[{"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1482"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1482"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.co.jp\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1482"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}