\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(a\\) \u304c\u6b63\u306e\u5b9f\u6570\u306e\u3068\u304d \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1+a^n \\right)^{\\frac{1}{n}}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u5b9a\u7a4d\u5206 \\(\\displaystyle\\int _ 1^{\\sqrt{3}} \\dfrac{1}{x^2} \\log \\sqrt{1+x^2} \\, dx\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(I(n) =\\left( 1+a^n \\right)^{\\frac{1}{n}}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\log I(n) =\\dfrac{\\log ( 1+a^n )}{n}\r\n\\]\r\n\\(a\\) \u306e\u5024\u306b\u3088\u3063\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n 1*<\/strong>\u3000\\(0 \\lt a \\lt 1\\) \u306e\u3068\u304d 2*<\/strong>\u3000\\(a=1\\) \u306e\u3068\u304d 3*<\/strong>\u3000\\(a \\gt 1\\) \u306e\u3068\u304d (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u5024\u3092 \\(J\\) \u3068\u304a\u304f.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n\\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, \\(1+a^n \\rightarrow 1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\log I(n) & = 0 \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\lim _ {n \\rightarrow \\infty} I(n) & = 1\r\n\\end{align}\\]<\/li>\r\n
\r\n\\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, \\(1+a^n \\rightarrow 2\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\log I(n) & = 0 \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\lim _ {n \\rightarrow \\infty} I(n) & = 1\r\n\\end{align}\\]<\/li>\r\n
\r\n\\(n \\gt 1\\) \u3067\u3042\u308c\u3070, \\(a^n \\lt 1+a^n \\lt a^{n+1}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{\\log a^n}{n} & \\lt \\log I(n) \\lt \\dfrac{\\log a^{n+1}}{n} \\\\\r\n\\text{\u2234} \\quad \\log a & \\lt \\log I(n) \\lt \\left( 1 +\\dfrac{1}{n} \\right) \\log a \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(n \\rightarrow \\infty\\) \u306e\u3068\u304d, [1] \u306e\u7b2c \\(1\\) \u8fba\u3068\u7b2c \\(3\\) \u8fba\u306f\u3068\u3082\u306b \\(\\log a\\) \u306b\u53ce\u675f\u3059\u308b\u306e\u3067, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\log I(n) & = \\log a \\\\\r\n\\text{\u2234} \\quad \\displaystyle\\lim _ {n \\rightarrow \\infty} I(n) & = a\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} I(n) = \\underline{\\left\\{ \\begin{array}{ll} 0 & \\ \\left( 0 \\lt a \\leqq 1 \\text{\u306e\u3068\u304d} \\right) \\\\ a & \\ \\left( a \\gt 1 \\text{\u306e\u3068\u304d} \\right) \\end{array} \\right.}\r\n\\]<\/li>\r\n<\/ol>\r\n
\r\n\\(f(x) =\\log \\sqrt{1+x^2}\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(x) =\\dfrac{1}{\\sqrt{1+x^2}} \\cdot \\dfrac{2x}{2 \\sqrt{1+x^2}} =\\dfrac{x}{1+x^2}\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068, \u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nJ & =\\left[ -\\dfrac{1}{x} \\log \\sqrt{1+x^2} \\right] _ 1^{\\sqrt{3}} +\\displaystyle\\int _ 1^{\\sqrt{3}} \\dfrac{1}{1+x^2} \\, dx \\\\\r\n& = \\dfrac{\\log 2}{2} -\\dfrac{\\log 2}{\\sqrt{3}} +\\underline{\\displaystyle\\int _ 1^{\\sqrt{3}} \\dfrac{1}{1+x^2} \\, dx} _ {[1]}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 [1] \u306b\u3064\u3044\u3066, \\(x= \\tan \\theta\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\dfrac{dx}{d \\theta} =\\dfrac{1}{\\cos^2 \\theta}\r\n\\]\r\n\u307e\u305f \\(x \\ : \\ 1 \\rightarrow \\sqrt{3}\\) \u306e\u3068\u304d, \\(\\theta \\ : \\ \\dfrac{\\pi}{4} \\rightarrow \\dfrac{\\pi}{3}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n[1] & = \\displaystyle\\int _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{3}} \\dfrac{1}{1+\\tan^2} \\cdot \\dfrac{d \\theta}{\\cos^2 \\theta} \\\\\r\n& = \\displaystyle\\int _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{3}} d \\theta =\\left[ \\theta \\right] _ {\\frac{\\pi}{4}}^{\\frac{\\pi}{3}} =\\dfrac{\\pi}{12}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nJ =\\underline{\\dfrac{\\pi}{12} +\\left( \\dfrac{1}{2} -\\dfrac{\\sqrt{3}}{3} \\right) \\log 2 }\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u5404\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(a\\) \u304c\u6b63\u306e\u5b9f\u6570\u306e\u3068\u304d \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1+a^n \\right)^{\\frac{1}{n}}\\) … \u7d9a\u304d\u3092\u8aad\u3080