\\(n\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong><\/p>\r\n \\(I _ n = n \\displaystyle\\int _ {-1}^1 g(nx) f(x) \\, dx\\) \u3068\u304a\u304f. (2)<\/strong><\/p>\r\n \\(J _ n = n^2 \\displaystyle\\int _ {-1}^1 h(nx) \\log ( 1 +e^{x+1} ) \\, dx\\) \u3068\u304a\u304f.
\r\n\\(nx = t\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nn \\, dx = dt , \\quad \\begin{array}{c|ccc} x & -1 & \\rightarrow & 1 \\\\ \\hline t & -n & \\rightarrow & n \\end{array}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nI _ n & = \\displaystyle\\int _ {-n}^n g(t) f \\left( \\dfrac{t}{n} \\right) \\, dt \\\\\r\n& = \\displaystyle\\int _ {-1}^1 g(t) f \\left( \\dfrac{t}{n} \\right) \\, dt \\quad ( \\ \\text{\u2235} \\ g(x) \\text{\u306e\u5b9a\u7fa9} \\ )\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\r\n\\displaystyle\\int _ {-1}^1 g(t) \\, dt = \\dfrac{1}{2} \\left[ \\dfrac{\\sin ( \\pi t )}{\\pi} +t \\right] _ {-1}^1 = 1\r\n\\]\r\n\u307e\u305f, \\(-1 \\leqq t \\leqq 1\\) \u306e\u3068\u304d, \\(p \\leqq f \\left( \\dfrac{t}{n} \\right) \\leqq q\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\np \\displaystyle\\int _ {-1}^1 g(t) \\, dt & \\leqq I _ n \\leqq q \\displaystyle\\int _ {-1}^1 g(t) \\, dt \\\\\r\n\\text{\u2234} \\quad p & \\leqq I _ n \\leqq q\r\n\\end{align}\\]\r\n
\r\n\\(|x| \\lt 1\\) \u306b\u304a\u3044\u3066\r\n\\[\r\ng'(x) = -\\dfrac{\\pi}{2} \\sin ( \\pi x ) = h(x)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\displaystyle\\int h(x) \\, dx = g(x) +C \\quad ( \\ C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} \\ )\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nJ _ n & = n^2 \\left[ \\dfrac{g(nx)}{n} \\log ( 1 +e^{x+1} )\\right] _ {-1}^1 -n \\displaystyle\\int _ {-1}^1 g(nx) \\cdot \\dfrac{e^{x+1}}{1 +e^{x+1}} \\, dx \\\\\r\n& = -n \\displaystyle\\int _ {-1}^1 g(nx) \\cdot \\underline{\\dfrac{e^{x+1}}{1 +e^{x+1}}} _ {[1]} \\, dx \\quad ( \\ \\text{\u2235} \\ g(n) = g(-n) = 0 \\ )\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, [1] \u3092 \\(f(x)\\) \u3068\u304a\u3051\u3070, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n-q \\leqq J _ n \\leqq -p \\quad ... [2]\r\n\\]\r\n\\(f(x)\\) \u306f\r\n\\[\r\nf(x) = 1 -\\dfrac{1}{1 +e^{x+1}}\r\n\\]\r\n\u3088\u308a, \u5358\u8abf\u5897\u52a0\u95a2\u6570\u306a\u306e\u3067, \\(|x| \\leqq \\dfrac{1}{n}\\) \u306b\u304a\u3051\u308b\u6700\u5c0f\u5024 \\(p\\) , \u6700\u5927\u5024 \\(q\\) \u306f\r\n\\[\\begin{align}\r\np & = 1 -\\dfrac{1}{1 +e^{-\\frac{1}{n} +1}} , \\\\\r\nq & = 1 -\\dfrac{1}{1 +e^{\\frac{1}{n} +1}}\r\n\\end{align}\\]\r\n\\(n \\rightarrow \\infty\\) \u306e\u6975\u9650\u3092\u8003\u3048\u308b\u3068\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} p = \\displaystyle\\lim _ {n \\rightarrow \\infty} q = 1 -\\dfrac{1}{1+e} = \\dfrac{e}{1+e}\r\n\\]\r\n\u3088\u3063\u3066, [2] \u3088\u308a, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u304b\u3089\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} J _ n = \\underline{-\\dfrac{e}{1+e}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u95a2\u6570 \\(g(x)\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u3081\u308b. \\[ g(x) = \\left\\{ \\begin{array}{ll} \\dfrac{\\cos ( \\pi x … \u7d9a\u304d\u3092\u8aad\u3080