\u6570\u5217 \\(\\left\\{ a _ n \\right\\} , \\left\\{ b _ n \\right\\}\\) \u3092\r\n\\[\\begin{align}\r\na _ n & = \\displaystyle\\int _ {-\\frac{\\pi}{6}}^{\\frac{\\pi}{6}} e^{n \\sin \\theta} \\, d \\theta , \\\\\r\nb _ n & = \\displaystyle\\int _ {-\\frac{\\pi}{6}}^{\\frac{\\pi}{6}} e^{n \\sin \\theta} \\cos \\theta \\, d \\theta \\quad ( n = 1, 2, 3, \\cdots ) \\ .\r\n\\end{align}\\]\r\n\u3067\u5b9a\u3081\u308b. \u305f\u3060\u3057, \\(e\\) \u306f\u81ea\u7136\u5bfe\u6570\u306e\u5e95\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u4e00\u822c\u9805 \\(\\left\\{ b _ n \\right\\}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u3059\u3079\u3066\u306e \\(n\\) \u306b\u3064\u3044\u3066, \\(b _ n \\leqq a _ n \\leqq \\dfrac{2}{\\sqrt{3}} b _ n\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{n} \\log \\left( na _ n \\right)\\) \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \u5bfe\u6570\u306f\u81ea\u7136\u5bfe\u6570\u3068\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\r\ne^{n \\sin \\theta} \\cos \\theta = \\dfrac{1}{n} \\left( e^{n \\sin \\theta} \\right)' \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nb _ n & = \\dfrac{1}{n} \\left[ e^{n \\sin \\theta} \\right] _ {-\\frac{\\pi}{6}}^{\\frac{\\pi}{6}} \\\\\r\n& = \\underline{\\dfrac{1}{n} \\left( e^{\\frac{n}{2}} -e^{-\\frac{n}{2}} \\right)} \\ .\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(-\\dfrac{\\pi}{6} \\leqq \\theta \\leqq \\dfrac{\\pi}{6}\\) \u306b\u304a\u3044\u3066\r\n\\[\r\n\\dfrac{\\sqrt{3}}{2} \\leqq \\cos \\theta \\leqq 1 , \\ e^{n \\sin \\theta} \\gt 0 \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\dfrac{\\sqrt{3}}{2} e^{n \\sin \\theta} \\leqq e^{n \\sin \\theta} \\cos \\theta \\leqq e^{n \\sin \\theta} \\ .\r\n\\]\r\n\u3053\u306e\u533a\u9593\u3067\u8fba\u3005\u3092\u7a4d\u5206\u3059\u308c\u3070\r\n\\[\\begin{gather}\r\n\\dfrac{\\sqrt{3}}{2} a _ n \\leqq b _ n \\leqq a _ n \\\\\r\n\\text{\u2234} \\quad b _ n \\leqq a _ n \\leqq \\dfrac{2}{\\sqrt{3}} a _ n \\ .\r\n\\end{gather}\\]\r\n (3)<\/strong><\/p>\r\n (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\dfrac{1}{n} \\log \\left( nb _ n \\right) \\leqq \\dfrac{1}{n} \\log \\left( na _ n \\right) \\leqq \\dfrac{1}{n} \\log \\left( \\dfrac{2}{\\sqrt{3}}nb _ n \\right) \\quad ... [1] \\ .\r\n\\]\r\n\u3053\u3053\u3067 (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070, [1] \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n( \\text{\u5de6\u8fba} ) & = \\dfrac{1}{n} \\log e^{\\frac{n}{2}} \\left( 1 -e^{-n} \\right) \\\\\r\n& = \\dfrac{1}{2} +\\dfrac{1}{n} \\log \\left( 1 -e^{-n} \\right) \\\\\r\n& \\rightarrow \\dfrac{1}{2} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} ) \\ , \\\\\r\n( \\text{\u53f3\u8fba} ) & = \\dfrac{1}{n} \\log \\dfrac{2}{\\sqrt{3}} e^{\\frac{n}{2}} \\left( 1 -e^{-n} \\right) \\\\\r\n& = \\dfrac{1}{2} +\\dfrac{1}{n} \\log \\left( 1 -e^{-n} \\right) +\\dfrac{1}{n} \\log \\dfrac{2}{\\sqrt{3}} \\\\\r\n& \\rightarrow \\dfrac{1}{2} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} ) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{n} \\log \\left( na _ n \\right) = \\underline{\\dfrac{1}{2}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6570\u5217 \\(\\left\\{ a _ n \\right\\} , \\left\\{ b _ n \\right\\}\\) \u3092 \\[\\begin{align} a _ n & = \\displaystyle\\int _ {-\\frac … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n