\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u95a2\u6570 \\(f(x) = \\dfrac{\\log (1-x)}{x}\\) \u306f \\(0 \\lt x \\lt 1\\) \u306e\u7bc4\u56f2\u3067\u6e1b\u5c11\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u6975\u9650\u5024\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{n} \\textstyle\\sum\\limits _ {k=1}^{n} \\dfrac{1}{\\tan \\left( \\dfrac{(n+k) \\pi}{6n} \\right)}\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\nf'(x) & = -\\dfrac{1}{1-x} \\cdot \\dfrac{1}{x} +\\log ( 1-x ) \\left( -\\dfrac{1}{x^2} \\right) \\\\\r\n& = -\\dfrac{x +(1-x) \\log ( 1-x )}{x^2 ( 1-x )}\r\n\\end{align}\\]\r\n\u5206\u5b50\u3092 \\(g(x)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\ng'(x) & = 1 -\\log ( 1-x ) +( 1-x ) \\left( -\\dfrac{1}{1-x} \\right) \\\\\r\n& = -\\log ( 1-x ) \\gt 0 \\quad ( \\ \\text{\u2235} \\ 1-x \\lt 1 \\ )\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(g(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u3067\r\n\\[\r\ng(x) \\gt g(0) = 0\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(1-x \\gt 0\\) \u306b\u3082\u6ce8\u610f\u3059\u308c\u3070\r\n\\[\r\nf'(x) \\lt 0\r\n\\]\r\n\u3088\u3063\u3066, \\(0 \\lt x \\lt 1\\) \u306b\u304a\u3044\u3066, \\(f(x)\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3059\u308b.<\/p>\r\n (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u5024\u3092 \\(S\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{n} \\textstyle\\sum\\limits _ {k=1}^{n} \\dfrac{1}{\\tan \\dfrac{\\pi}{6} \\left( 1 +\\dfrac{k}{n} \\right)} \\\\\r\n& = \\displaystyle\\int _ {0}^{1} \\dfrac{1}{\\tan \\dfrac{\\pi}{6} ( 1+x )} \\, dx \\\\\r\n& = \\dfrac{6}{\\pi} \\displaystyle\\int _ {0}^{1} \\dfrac{\\left\\{ \\sin \\dfrac{\\pi}{6} ( 1+x ) \\right\\}'}{\\sin \\dfrac{\\pi}{6} ( 1+x )} \\, dx \\\\\r\n& = \\dfrac{6}{\\pi} \\left[ \\log \\left| \\sin \\dfrac{\\pi}{6} ( 1+x ) \\right| \\right] _ {0}^{1} \\\\\r\n& = \\dfrac{6}{\\pi} \\left( \\log \\sin \\dfrac{\\pi}{3} -\\log \\sin \\dfrac {\\pi}{6} \\right) \\\\\r\n& = \\dfrac{6}{\\pi} \\log \\dfrac{\\dfrac{\\sqrt{3}}{2}}{\\dfrac{1}{2}} \\\\\r\n& = \\underline{\\dfrac{3}{\\pi} \\log 3}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u95a2\u6570 \\(f(x) = \\dfrac{\\log (1-x)}{x}\\) \u306f \\(0 \\lt x \\lt 1\\) \u306e\u7bc4\u56f2\u3067\u6e1b\u5c11\u3059\u308b\u3053\u3068\u3092\u793a\u305b. (2)\u3000\u6975\u9650\u5024 \\[ \\displaystyle\\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n