{"id":610,"date":"2013-02-11T09:50:39","date_gmt":"2013-02-11T00:50:39","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=610"},"modified":"2021-03-24T09:47:20","modified_gmt":"2021-03-24T00:47:20","slug":"kyr200702","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr200702\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb\u4e592007\uff1a\u7b2c2\u554f"},"content":{"rendered":"
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\\(x , y\\) \u3092\u76f8\u7570\u306a\u308b\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092\r\n\\[\r\na _ 1 = 0 , \\ a _ {n+1} = x a _ n +y^{n+1} \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n\\) \u304c\u6709\u9650\u306e\u5024\u306b\u53ce\u675f\u3059\u308b\u3088\u3046\u306a\u5ea7\u6a19\u5e73\u9762\u4e0a\u306e\u70b9 \\(( x , y )\\) \u306e\u7bc4\u56f2\u3092\u56f3\u793a\u305b\u3088.<\/p>\r\n

\r\n\r\n

\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

\u4e0e\u3048\u3089\u308c\u305f\u6f38\u5316\u5f0f\u3092\u4e21\u8fba \\(x^{n+1}\\) \u3067\u5272\u308b\u3068\r\n\\[\r\n\\dfrac{a _ {n+1}}{x^{n+1}} = \\dfrac{a _ n}{x^n} +\\left( \\dfrac{y}{x} \\right)^{n+1}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n \\geqq 2\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{a _ n}{x^n} & = \\dfrac{a _ 1}{x} +\\displaystyle\\sum _ {k=1}^{n-1} \\left( \\dfrac{y}{x} \\right)^{k+1} \\\\\r\n& = \\dfrac{y^2}{x^2} \\cdot \\dfrac{1 -\\left( \\frac{y}{x} \\right)^{n-1}}{1 -\\frac{y}{x}} \\quad ( \\ \\text{\u2235} \\ \\dfrac{y}{x} \\neq 1 \\ ) \\\\\r\n& = \\dfrac{y^2 \\left\\{ 1 -\\left( \\frac{y}{x} \\right)^{n-1} \\right\\}}{x(x-y)} \\\\\r\n\\text{\u2234} \\quad a _ n & = \\dfrac{y^2 \\left( x^{n-1} -y^{n-1} \\right)}{x-y}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(b _ n = x^{n-1} -y^{n-1}\\) \u3068\u304a\u3044\u3066, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} b _ n\\) \u304c\u6709\u9650\u306e\u5024\u306b\u53ce\u675f\u3059\u308b\u6761\u4ef6\u3092\u8003\u3048\u308c\u3070\u3088\u3044.<\/p>\r\n

    \r\n

  1. 1*<\/strong>\u3000\\(\\dfrac{y}{x} \\lt 1\\) \u3059\u306a\u308f\u3061 \\(0 \\lt y \\lt x\\) \u306e\u3068\u304d\r\n\\[\r\nb _ n = x^{n-1} \\left\\{ 1 -\\left( \\dfrac{y}{x} \\right)^{n-1} \\right\\}\r\n\\]\r\n\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\dfrac{y}{x} \\right)^{n-1}\\) \u306a\u306e\u3067, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} x^{n-1}\\) \u304c\u6709\u9650\u306e\u5024\u306b\u53ce\u675f\u3059\u308c\u3070\u3088\u304f\r\n\\[\r\n0 \\lt y \\lt x \\leqq 1\r\n\\]<\/li>\r\n

  2. 2*<\/strong>\u3000\\(\\dfrac{x}{y} \\lt 1\\) \u3059\u306a\u308f\u3061 \\(0 \\lt x \\lt y\\) \u306e\u3068\u304d\r\n\\[\r\nb _ n = y^{n-1} \\left\\{ 1 -\\left( \\dfrac{x}{y} \\right)^{n-1} \\right\\}\r\n\\]\r\n\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\dfrac{x}{y} \\right)^{n-1}\\) \u306a\u306e\u3067, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} y^{n-1}\\) \u304c\u6709\u9650\u306e\u5024\u306b\u53ce\u675f\u3059\u308c\u3070\u3088\u304f\r\n\\[\r\n0 \\lt x \\lt y \\leqq 1\r\n\\]<\/li>\r\n<\/ol>\r\n

    \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u7bc4\u56f2\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u305f\u3060\u3057, \u25cb, \\(x\\) \u8ef8, \\(y\\) \u8ef8, \u76f4\u7dda \\(y=x\\) \u4e0a\u306e\u70b9\u306f\u9664\u304f\uff09.<\/p>\r\n\"kyoto_r_2007_02_01\"\r\n","protected":false},"excerpt":{"rendered":"\\(x , y\\) \u3092\u76f8\u7570\u306a\u308b\u6b63\u306e\u5b9f\u6570\u3068\u3059\u308b. \u6570\u5217 \\(\\{ a _ n \\}\\) \u3092 \\[ a _ 1 = 0 , \\ a _ {n+1} = x a _ n +y^{n+1} \\quad ( n = 1, 2, 3 … \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":[87],"tags":[140,109],"class_list":["post-610","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2007","tag-kyoto_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/610","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=610"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/610\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=610"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=610"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=610"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}