\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \u95a2\u6570\r\n\\[\r\nF _ n(x) = \\displaystyle\\int _ x^{2x} e^{-t^n} \\, dt \\quad ( x \\geqq 0 )\r\n\\]\r\n\u3092\u8003\u3048\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u95a2\u6570 \\(F _ n(x) \\ ( x \\geqq 0 )\\) \u306f\u305f\u3060\u4e00\u3064\u306e\u70b9\u3067\u6700\u5927\u5024\u3092\u3068\u308b\u3053\u3068\u3092\u793a\u3057, \\(F _ n(x)\\) \u304c\u6700\u5927\u3068\u306a\u308b\u3088\u3046\u306a \\(x\\) \u306e\u5024 \\(a _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000(1)<\/strong> \u3067\u6c42\u3081\u305f \\(a _ n\\) \u306b\u5bfe\u3057, \u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\log a _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(f _ n(t) = \\displaystyle\\int e^{-t^n} \\, dt\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nf' _ n(t) = e^{-t^n}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \\(F _ n(x) = f _ n(2x) -f _ n(x)\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nF' _ n(x) & = 2f' _ n(2x) -f' _ n(x) = 2 e^{-(2x)^n} -e^{-x^n} \\\\\r\n& = 2 \\left( e^{-x^n} \\right)^{2^n} -e^{-x^n}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(X =e^{-x^n}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nF' _ n(x) = 2 X^{2^n} -X = X \\left( 2X^{2^n-1} -1 \\right)\r\n\\]\r\n\\(X = e^{-x^n}\\) \u306f, \\(n\\) \u304c\u81ea\u7136\u6570\u306a\u306e\u3067, \\(x \\geqq 0\\) \u306b\u304a\u3044\u3066\u5358\u8abf\u6e1b\u5c11\u3057, \\(\\displaystyle\\lim _ {x \\rightarrow \\infty} e^{-x^n} =0\\) \u306a\u306e\u3067\r\n\\[\r\n0 \\lt X \\leqq e^{-0^n} =1\r\n\\]\r\n\u3053\u306e\u7bc4\u56f2\u3067, \\(F' _ n(x) =0\\) \u3092\u89e3\u304f\u3068\r\n\\[\\begin{align}\r\nX^{2^n-1} & = \\dfrac{1}{2} \\\\\r\nX & = 2^{- \\frac{1}{2^n-1}} \\\\\r\n\\text{\u2234} \\quad e^{-x^n} & = 2^{- \\frac{1}{2^n-1}} \\\\\r\n\\text{\u2234} \\quad -x^n & = -\\dfrac{\\log 2}{2^n-1} \\\\\r\n\\text{\u2234} \\quad x & = \\sqrt[n]{\\dfrac{\\log 2}{2^n-1}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(F _ n(x)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|cccc} x & 0 & \\cdots & \\sqrt[n]{\\dfrac{\\log 2}{2^n-1}} & \\cdots \\\\ \\hline F _ n'(x) & & + & 0 & - \\\\ \\hline F _ n(x) & & \\nearrow & \\text{\u6700\u5927} & \\searrow \\end{array}\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\r\na _ n = \\underline{\\sqrt[n]{\\dfrac{\\log 2}{2^n-1}}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\log a _ n & = \\dfrac{1}{n} \\left\\{ \\log 2 -\\log ( 2^n-1 ) \\right\\} \\\\\r\n& = \\dfrac{1}{n} \\left\\{ \\log 2 -\\log 2^n \\left( 1 -\\dfrac{1}{2^n} \\right) \\right\\} \\\\\r\n& = \\dfrac{\\log 2}{n} -\\log 2 -\\dfrac{1}{n} \\cdot \\log \\left( 1 -\\dfrac{1}{2^n} \\right) \\\\\r\n& \\rightarrow 0 -\\log 2 -0 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} ) \\\\\r\n& = -\\log 2\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n = \\underline{- \\log 2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \u95a2\u6570 \\[ F _ n(x) = \\displaystyle\\int _ x^{2x} e^{-t^n} \\, dt \\quad ( x \\geqq 0 ) \\] \u3092\u8003\u3048\u308b. (1)\u3000\u95a2\u6570 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n