{"id":1971,"date":"2021-11-16T08:14:51","date_gmt":"2021-11-15T23:14:51","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1971"},"modified":"2021-11-16T08:14:51","modified_gmt":"2021-11-15T23:14:51","slug":"kyr202103","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr202103\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb2021\uff1a\u7b2c3\u554f"},"content":{"rendered":"<hr \/>\n<p>\u7121\u9650\u7d1a\u6570 \\(\\textstyle\\sum\\limits _ {n=0}^{\\infty} \\left( \\dfrac{1}{2} \\right)^n \\cos \\dfrac{n \\pi}{6}\\) \u306e\u548c\u3092\u6c42\u3081\u3088.<\/p>\r\n<hr \/>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p>\\(\\alpha = \\cos \\dfrac{\\pi}{6} +i \\sin \\dfrac{\\pi}{6} = \\dfrac{\\sqrt{3}}{2} +\\dfrac{i}{2}\\) \u3068\u304a\u304f\u3068<br \/>\r\n\\(\\alpha^n = \\cos \\dfrac{n \\pi}{6} +i \\sin \\dfrac{n \\pi}{6}\\) \u306a\u306e\u3067\r\n\\[\r\n\\left( \\dfrac{1}{2} \\right)^n \\cos \\dfrac{n \\pi}{6} = \\dfrac{1}{2} \\left\\{ \\left( \\dfrac{\\alpha}{2} \\right)^n +\\left( \\dfrac{\\overline{\\alpha}}{2} \\right)^n \\right\\}\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u548c \\(S\\) \u306f\r\n\\[\r\nS = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{2} \\left\\{ \\dfrac{1 -\\left( \\dfrac{\\alpha}{2} \\right)^{n+1}}{1 -\\dfrac{\\alpha}{2}}\r\n+\\dfrac{1 -\\left( \\dfrac{\\overline{\\alpha}}{2} \\right)^{n+1}}{1 -\\dfrac{\\overline{\\alpha}}{2}} \\right\\}\r\n\\]\r\n\\(\\left| \\dfrac{\\alpha}{2} \\right| = \\left| \\dfrac{\\overline{\\alpha}}{2} \\right| = \\dfrac{1}{2}\\) \u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left| \\dfrac{\\alpha}{2} \\right|^{n+1} = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left| \\dfrac{\\overline{\\alpha}}{2} \\right|^{n+1} = 0\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\dfrac{\\alpha}{2} \\right)^{n+1} = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( \\dfrac{\\overline{\\alpha}}{2} \\right)^{n+1} = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nS & = \\dfrac{1}{2} \\left( \\dfrac{1}{1 -\\dfrac{\\alpha}{2}}\r\n+\\dfrac{1}{1 -\\dfrac{\\overline{\\alpha}}{2}} \\right) \\\\\r\n& = \\dfrac{1}{2 -\\alpha} +\\dfrac{1}{2 -\\overline{\\alpha}} \\\\\r\n& = \\dfrac{4 -\\alpha -\\overline{\\alpha}}{| 2 -\\alpha |^2}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n| 2 -\\alpha |^2 & = \\left( 2 -\\dfrac{\\sqrt{3}}{2} \\right)^2 +\\left( \\dfrac{1}{2} \\right)^2 \\\\\r\n& = 4 +\\dfrac{3}{4} -2\\sqrt{3} +\\dfrac{1}{4} \\\\\r\n& = 5 -2 \\sqrt{3} \\ , \\\\\r\n4 -\\alpha -\\overline{\\alpha} & = 4 -\\sqrt{3}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS & = \\dfrac{4 -\\sqrt{3}}{5 -2 \\sqrt{3}} \\\\\r\n& = \\dfrac{\\left( 4 -\\sqrt{3} \\right) \\left( 5 +2 \\sqrt{3} \\right)}{13} \\\\\r\n& = \\underline{\\dfrac{14 +3 \\sqrt{3}}{13}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u7121\u9650\u7d1a\u6570 \\(\\textstyle\\sum\\limits _ {n=0}^{\\infty} \\left( \\dfrac{1}{2} \\right)^n \\cos \\dfrac{n \\pi}{6}\\) \u306e\u548c\u3092\u6c42\u3081\u3088.","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":[171],"tags":[140,165],"class_list":["post-1971","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2021","tag-kyoto_r","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1971","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=1971"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1971\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1971"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1971"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1971"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}