{"id":432,"date":"2012-06-20T00:34:22","date_gmt":"2012-06-19T15:34:22","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=432"},"modified":"2021-09-30T08:40:02","modified_gmt":"2021-09-29T23:40:02","slug":"kbr201202","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201202\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2012\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u66f2\u7dda \\(C : \\ y =\\dfrac{1}{x+2} \\ ( x \\gt -2 )\\) \u3092\u8003\u3048\u308b.\r\n\u66f2\u7dda \\(C\\) \u4e0a\u306e\u70b9 \\(P _ 1 \\left( 0 , \\dfrac{1}{2} \\right)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(\\ell _ 1\\) \u3068\u3057, \\(\\ell _ 1\\) \u3068 \\(x\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u3092 \\(Q _ 1\\) , \u70b9 \\(Q _ 1\\) \u3092\u901a\u308a \\(x\\) \u8ef8\u3068\u5782\u76f4\u306a\u76f4\u7dda\u3068\u66f2\u7dda \\(C\\) \u3068\u306e\u4ea4\u70b9\u3092 \\(P _ 2\\) \u3068\u304a\u304f. \u4ee5\u4e0b\u540c\u69d8\u306b, \u81ea\u7136\u6570 \\(n \\ ( n \\geqq 2 )\\) \u306b\u5bfe\u3057\u3066, \u70b9 \\(P _ n\\) \u306b\u304a\u3051\u308b\u63a5\u7dda\u3092 \\(\\ell _ n\\) \u3068\u3057, \\(\\ell _ n\\) \u3068 \\(x\\) \u8ef8\u3068\u306e\u4ea4\u70b9\u3092 \\(Q _ n\\) , \u70b9 \\(Q _ n\\) \u3092\u901a\u308a \\(x\\) \u8ef8\u3068\u5782\u76f4\u306a\u76f4\u7dda\u3068\u66f2\u7dda \\(C\\) \u3068\u306e\u4ea4\u70b9\u3092 \\(P _ {n+1}\\) \u3068\u304a\u304f.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\ell _ 1\\) \u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(P _ n\\) \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(x _ n \\ ( n \\geqq 1 )\\) \u3068\u3059\u308b. \\(x _ {n+1}\\) \u3092 \\(x _ n\\) \u3092\u7528\u3044\u3066\u8868\u3057, \\(x _ n\\) \u3092 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\ell _ n\\) , \\(x\\) \u8ef8, \\(y\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u4e09\u89d2\u5f62\u306e\u9762\u7a4d \\(S _ n\\) \u3092\u6c42\u3081, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(C\\) \u306e\u5f0f\u3088\u308a\r\n\\[\r\ny' = -\\dfrac{1}{(x+2)^2}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\ell _ 1 : \\ \\underline{y =-\\dfrac{x}{4} +\\dfrac{1}{2}}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\[\r\n\\ell _ n : \\ y = -\\dfrac{x-x _ n}{(x _ n+2)^2} +\\dfrac{1}{x _ n+2}\r\n\\]\r\n\\(y = 0\\) \u3092\u89e3\u304f\u3068\r\n\\[\\begin{gather}\r\n-\\dfrac{x-x _ n}{x _ n+2}+1 = 0 \\\\\r\n\\text{\u2234} \\quad x = 2x _ n+2\r\n\\end{gather}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nx _ {n+1} & = 2x _ n+2 \\\\\r\nx _ {n+1}+2 & = 2(x _ n+2) \\\\\r\n\\text{\u2234} \\quad x _ n+2 & = 2^{n-1} (x _ 1+2) = 2^n\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nx _ n =\\underline{2^n-2}\r\n\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\(\\ell _ n\\) \u306b\u3064\u3044\u3066, \\(x\\) \u5207\u7247\u306f\r\n\\[\r\nx _ {n+1} = 2(x _ n+1)\r\n\\]\r\n\u307e\u305f, \\(y\\) \u5207\u7247\u306f\r\n\\[\r\n\\dfrac{x _ n}{(x _ n+2)^2} +\\dfrac{1}{x _ n+2} = \\dfrac{2 (x _ n+1)}{(x _ n+2)^2}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS _ n & = \\dfrac{1}{2} \\cdot 2(x _ n+1) \\cdot \\dfrac{2 (x _ n+1)}{(x _ n+2)^2} \\\\\r\n& =\\underline{\\dfrac{( 2^n-1 )^2}{2^{2n-1}}} \\\\\r\n& = 2 \\left( 1 +\\dfrac{1}{2^n} \\right)^2 \\\\\r\n& \\rightarrow 2 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n = \\underline{2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u66f2\u7dda \\(C : \\ y =\\dfrac{1}{x+2} \\ ( x \\gt -2 )\\) \u3092\u8003\u3048\u308b. \u66f2\u7dda \\(C\\) \u4e0a\u306e\u70b9 \\(P _ 1 \\left( 0 , \\dfrac{1}{2} \\right)\\) \u306b\u304a\u3051 … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[65],"tags":[144,68],"class_list":["post-432","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2012","tag-tsukuba_r","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/432","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=432"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/432\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=432"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=432"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=432"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}