{"id":1385,"date":"2017-02-25T01:15:17","date_gmt":"2017-02-24T16:15:17","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1385"},"modified":"2021-03-18T16:13:15","modified_gmt":"2021-03-18T07:13:15","slug":"kyr201601","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr201601\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2016\uff1a\u7b2c1\u554f"},"content":{"rendered":"<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b\u3068\u304d, \u95a2\u6570\r\n\\[\r\nf _ n ( \\theta ) = ( 1 +\\cos \\theta ) \\sin^{n-1} \\theta\r\n\\]\r\n\u306e \\(0 \\leqq \\theta \\leqq \\dfrac{\\pi}{2}\\) \u306b\u304a\u3051\u308b\u6700\u5927\u5024 \\(M _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( M _ n \\right)^n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr\/>\r\n\r\n    <div id='extLink1385'>\r\n        <p>\r\n            <a href='https:\/\/www.roundown.net\/nyushi\/kyr201601\/' name='ext1385' onclick=\"showHide(1385,'https:\/\/www.roundown.net\/nyushi\/kyr201601\/',this,'entry');return false;\">\u89e3\u7b54\u306f\u3053\u3061\u3089 &#187;<\/a>\r\n        <\/p>\r\n    <\/div>\r\n    <div id='extText1385' style='display: none'>\r\n        \r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\nf'( \\theta ) & = -\\sin^n +( 1 +\\cos \\theta ) (n-1) \\sin^{n-2} \\theta \\cos \\theta \\\\\r\n& = \\sin^{n-2} \\theta \\left\\{ \\cos^2 \\theta -1 +(n-1) \\cos \\theta ( 1 +\\cos \\theta ) \\right\\} \\\\\r\n& = \\sin^{n-2} \\theta \\left\\{ \\cos^2 \\theta +(n-1) \\cos \\theta -1 \\right\\} \\\\\r\n& = \\sin^{n-2} \\theta ( n \\cos \\theta -1 ) ( \\cos \\theta +1 )\r\n\\end{align}\\]\r\n\\(f'( \\theta ) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\n\\theta = 0 , \\ \\cos \\theta = \\dfrac{1}{n}\r\n\\]\r\n\\(\\cos \\theta _ n = \\dfrac{1}{n} \\ \\left( 0 \\leqq \\theta _ n \\leqq \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u3051\u3070, \\(f( \\theta )\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & 0 & \\cdots & \\theta _ n & \\cdots & \\dfrac{\\pi}{2} \\\\ \\hline f'( \\theta ) & 0 & + & 0 & - & \\\\ \\hline f( \\theta ) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\\\ \\end{array}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nM _ n & = f( \\theta _ n ) \\\\\r\n& = \\underline{\\left( 1 +\\dfrac{1}{n} \\right) \\left( \\sqrt{1 -\\dfrac{1}{n^2}} \\right)^{n-1}} \\\\\r\n\\end{align}\\]\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>\\[\\begin{align}\r\n\\left( M _ n \\right)^n & = \\left( 1 +\\dfrac{1}{n} \\right)^n \\left( 1 -\\dfrac{1}{n^2} \\right)^{\\frac{n^2 -n}{2}} \\\\\r\n& = \\left( 1 +\\dfrac{1}{n} \\right)^n \\left( 1 -\\dfrac{1}{n^2} \\right)^{-\\frac{1}{2} (-n^2)} \\left( 1 +\\dfrac{1}{n} \\right)^{-\\frac{1}{2} n} \\left( 1 -\\dfrac{1}{n} \\right)^{\\frac{1}{2} (-n)} \\\\\r\n& \\rightarrow e \\cdot e^{-\\frac{1}{2}} \\cdot e^{-\\frac{1}{2}} \\cdot e^{\\frac{1}{2}} \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} ) \\\\\r\n& = e^{\\frac{1}{2}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( M _ n \\right)^n = \\underline{e^{\\frac{1}{2}}}\r\n\\]\r\n\r\n        <p>\r\n            <a href='#ext1385' onclick=\"showHide(1385,0,this,'entry');return true;\">&#171; \u89e3\u7b54\u3092\u96a0\u3059 <\/a>\r\n        <\/p>\r\n    <\/div>","protected":false},"excerpt":{"rendered":"(1)\u3000\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b\u3068\u304d, \u95a2\u6570 \\[ f _ n ( \\theta ) = ( 1 +\\cos \\theta ) \\sin^{n-1} \\theta \\] \u306e \\(0 \\leqq \\t [&hellip;]","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":[151],"tags":[140,162],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1385"}],"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=1385"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1385\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1385"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1385"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1385"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}