{"id":34,"date":"2011-11-25T21:57:56","date_gmt":"2011-11-25T12:57:56","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=34"},"modified":"2021-09-24T17:54:35","modified_gmt":"2021-09-24T08:54:35","slug":"tok201101","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201101\/","title":{"rendered":"\u6771\u5de5\u59272011\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u3067\u884c\u5217 \\(\\left( \\begin{array}{cc} 1-n & 1 \\\\ -n( n+1 ) & n+2 \\end{array} \\right)\\) \u306e\u8868\u3059 \\(1\\) \u6b21\u5909\u63db\uff08\u79fb\u52d5\u3068\u3082\u3044\u3046\uff09\u3092 \\(f _ n\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u539f\u70b9 O \\(( 0 , 0 )\\) \u3092\u901a\u308b\u76f4\u7dda\u3067, \u305d\u306e\u76f4\u7dda\u4e0a\u306e\u3059\u3079\u3066\u306e\u70b9\u304c \\(f _ n\\) \u306b\u3088\u308a\u540c\u3058\u76f4\u7dda\u4e0a\u306b\u79fb\u3055\u308c\u308b\u3082\u306e\u304c \\(2\\) \u672c\u3042\u308b\u3053\u3068\u3092\u793a\u3057, \u3053\u306e \\(2\\) \u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u3067\u5f97\u3089\u308c\u305f \\(2\\) \u76f4\u7dda\u3068\u66f2\u7dda \\(y = x^2\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u56f3\u5f62\u306e\u9762\u7a4d \\(S _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(\\textstyle\\sum\\limits _ {n=1}^{\\infty} \\dfrac{1}{S _ n -\\frac{1}{6}}\\) \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

    \u539f\u70b9 O \u3092\u901a\u308b\u76f4\u7dda\u306f, \\(x = 0\\) \u307e\u305f\u306f \\(y = kx\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u3068\u8868\u305b\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(x = 0\\) \u306e\u3068\u304d, \u70b9P \\(( p , 0 )\\) \u304c \\(f _ n\\) \u306b\u3088\u3063\u3066\u79fb\u52d5\u3059\u308b\u70b9\u306f\r\n\\[\r\n\\left( \\begin{array}{cc} 1-n & 1 \\\\ -n( n+1 ) & n+2 \\end{array} \\right) \\left( \\begin{array}{c} p \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} p( n-1 ) \\\\ -pn( n+1 ) \\end{array} \\right)\r\n\\]\r\n\u3053\u306e\u70b9\u304c\u4efb\u610f\u306e \\(p\\) \u306b\u5bfe\u3057\u3066 \\(x = 0\\) \u4e0a\u306b\u3042\u308b\u3053\u3068\u306f\u306a\u3044\u306e\u3067\u4e0d\u9069.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(y = kx\\) \u306e\u3068\u304d\r\n\\[\r\n\\left( \\begin{array}{cc} 1-n & 1 \\\\ -n( n+1 ) & n+2 \\end{array} \\right) \\left( \\begin{array}{c} p \\\\ kp \\end{array} \\right) = \\left( \\begin{array}{c} p( 1-n+k ) \\\\ p \\left\\{ -n( n+1 ) +k( n+2 ) \\right\\} \\end{array} \\right)\r\n\\]\r\n\u3053\u306e\u70b9\u304c \\(y = kx\\) \u4e0a\u306b\u3042\u308b\u306e\u3067\r\n\\[\r\np \\left\\{ -n( n+1 ) +k( n+2 ) \\right\\} = k p( 1-n+k )\r\n\\]\r\n\u3053\u308c\u304c\u4efb\u610f\u306e \\(p\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u306e\u3067,\r\n\\[\\begin{align}\r\n-n( n+1 ) +k( n+2 ) & = k ( 1-n+k ) \\\\\r\nk^2 -( 2n+1 )k +n( n+1 ) & = 0 \\\\\r\n( k-n )( k-n-1 ) & = 0 \\\\\r\n\\text{\u2234} \\quad k & = n , n+1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(y = nx\\) \u3068 \\(y = (n+1)x\\) \u306e \\(2\\) \u672c\u304c\u6761\u4ef6\u3092\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a\u6761\u4ef6\u3092\u6e80\u305f\u3059\u76f4\u7dda\u306f \\(2\\) \u672c\u3042\u308a, \u305d\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\n\\underline{y = nx , \\ y = (n+1)x}\r\n\\]\r\n

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

      \\(y = nx , \\ y = (n+1)x\\) \u306f\u305d\u308c\u305e\u308c \\(y=x^2\\) \u3068 \\(2\\) \u3064\u306e\u4ea4\u70b9 \\(( 0 , 0 )\\) \u3068 \\(( n , n^2 )\\) , \\(( 0 , 0 )\\) \u3068 \\(( n+1 , (n+1)^2 )\\) \u3092\u3082\u3064.<\/p>\r\n\"\"\r\n

      \u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS _ n & = \\displaystyle\\int _ 0^{n+1} \\left\\{ x^2 -(n+1)x \\right\\} \\, dx -\\displaystyle\\int _ 0^n ( x^2 -nx ) \\, dx \\\\\r\n& = \\dfrac{1}{6} (n+1)^3 -\\dfrac{1}{6} n^3 \\\\\r\n& = \\underline{\\dfrac{1}{6} ( 3n^2 +3n +1 )}\r\n\\end{align}\\]\r\n

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

      \\[\r\n\\dfrac{1}{S _ n -\\dfrac{1}{6}}= \\dfrac{2}{n^2 +n} = 2 \\left( \\dfrac{1}{n} -\\dfrac{1}{n+1} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {n=1}^{\\infty} \\dfrac{1}{S _ n -\\frac{1}{6}} & = 2 \\displaystyle\\lim _ {n \\rightarrow \\infty} \\textstyle\\sum\\limits _ {k=1}^{n} \\left( \\dfrac{1}{k} -\\dfrac{1}{k+1} \\right) \\\\\r\n& = 2 \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1 -\\dfrac{1}{n+1} \\right) \\\\\r\n& = \\underline{2}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u3067\u884c\u5217 \\(\\left( \\begin{array}{cc} 1-n & 1 \\\\ -n( n+1 ) & n+2 \\end{array} \\right)\\) \u306e\u8868\u3059 \\ … \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":[26],"tags":[141,13],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-toko_2011","tag-toko","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/34","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=34"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/34\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=34"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=34"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=34"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}