{"id":801,"date":"2013-05-30T00:09:34","date_gmt":"2013-05-29T15:09:34","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=801"},"modified":"2021-09-24T17:41:08","modified_gmt":"2021-09-24T08:41:08","slug":"tok201304","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201304\/","title":{"rendered":"\u6771\u5de5\u59272013\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306e\u7bc4\u56f2\u306b\u304a\u3044\u3066 \\(\\sin 4nx \\geqq \\sin x\\) \u3092\u6e80\u305f\u3059 \\(x\\) \u306e\u533a\u9593\u306e\u9577\u3055\u306e\u7dcf\u548c\u3092 \\(S _ n\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n<hr>\r\n<!--more-->\r\n<h2>\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2013_04_01.png\" alt=\"toko_2013_04_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u4e0a\u56f3\u3088\u308a, \u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092\u307f\u305f\u3059 \\(x\\) \u306e\u7bc4\u56f2\u306f, \\(0\\) \u4ee5\u4e0a\u306e\u6574\u6570 \\(k\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nx & \\leqq 4nx -2k \\pi \\leqq \\pi -x \\\\\r\n2k \\pi & \\leqq (4n-1) x , \\ (4n+1) x \\leqq (2k+1) \\pi \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\dfrac{2k+1}{4n+1} & -\\dfrac{2k}{4n-1} \\\\\r\n& = \\dfrac{(2k+1)(4n-1) -2k (4n+1)}{(4n+1)(4n-1)} \\\\\r\n& = \\dfrac{(4n-1)-4k}{(4n+1)(4n-1)}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(n , k\\) \u304c\u3068\u3082\u306b\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066 [1] \u3092\u3068\u304f\u3068\r\n\\[\r\n\\left\\{\\begin{array}{ll} \\dfrac{2k}{4n-1} \\pi \\leqq x \\leqq \\dfrac{2k+1}{4n+1} \\pi & \\left( 0 \\leqq k \\leqq n-1 \\text{\u306e\u3068\u304d} \\right) \\\\ \\text{\u89e3\u306a\u3057} & \\left( k \\geqq n \\text{\u306e\u3068\u304d} \\right) \\end{array}\\right.\r\n\\]\r\n\u3053\u306e\u3046\u3061, \\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306b\u542b\u307e\u308c\u308b\u306e\u306f, \\(0 \\leqq k \\leqq n-1\\) \u306e\u3068\u304d\u3067\u3042\u308b.<br \/>\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS _ n & = \\textstyle\\sum\\limits _ {k=0}^{n-1} \\left( \\dfrac{2k+1}{4n+1} -\\dfrac{2k}{4n-1} \\right) \\pi \\\\\r\n& = \\dfrac{\\pi}{(4n+1)(4n-1)} \\textstyle\\sum\\limits _ {k=0}^{n-1} \\left\\{ (4n-1)-4k \\right\\} \\\\\r\n& = \\dfrac{\\pi \\left\\{ n(4n-1) -2 n(n-1) \\right\\}}{(4n+1)(4n-1)} \\\\\r\n& = \\dfrac{n (2n+1) \\pi}{(4n+1)(4n-1)} \\\\\r\n& = \\dfrac{\\left(2 +\\frac{1}{n} \\right) \\pi}{\\left( 4 +\\frac{1}{n} \\right) \\left( 4 -\\frac{1}{n} \\right)} \\\\\r\n& \\rightarrow \\dfrac{2 \\pi}{4 \\cdot 4} = \\dfrac{\\pi}{8} \\quad ( n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} S _ n = \\underline{\\dfrac{\\pi}{8}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057, \\(0 \\leqq x \\leqq \\dfrac{\\pi}{2}\\) \u306e\u7bc4\u56f2\u306b\u304a\u3044\u3066 \\(\\sin 4nx \\geqq \\sin x\\) \u3092\u6e80\u305f\u3059 \\(x\\) \u306e\u533a\u9593\u306e\u9577\u3055\u306e\u7dcf\u548c\u3092 \\(S &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201304\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[92],"tags":[141,111],"class_list":["post-801","post","type-post","status-publish","format-standard","hentry","category-toko_2013","tag-toko","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/801","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=801"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/801\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=801"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=801"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=801"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}