\\(a \\gt 0\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(n = 1, 2, 3, \\cdots\\) \u306b\u5bfe\u3057, \u5ea7\u6a19\u5e73\u9762\u306e \\(3\\) \u70b9\r\n\\[\r\n( 2n \\pi , 0 ) , \\ \\left( \\left( 2n +\\dfrac{1}{2} \\right) \\pi , \\dfrac{1}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} \\right) , \\ \\left( (2n+1) \\pi , 0 \\right)\r\n\\]\r\n\u3092\u9802\u70b9\u3068\u3059\u308b\u4e09\u89d2\u5f62\u306e\u9762\u7a4d\u3092 \\(A _ n\\) \u3068\u3057,\r\n\\[\r\nB _ n = \\displaystyle\\int _ {2n \\pi}^{(2n+1) \\pi} \\dfrac{\\sin x}{x^a} \\, dx , \\ C _ n = \\displaystyle\\int _ {2n \\pi}^{(2n+1) \\pi} \\dfrac{\\sin^2 x}{x^a} \\, dx\r\n\\]\r\n\u3068\u304a\u304f.<\/p>\r\n
(1)<\/strong>\u3000\\(n = 1, 2, 3, \\cdots\\) \u306b\u5bfe\u3057, \u6b21\u306e\u4e0d\u7b49\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.\r\n\\[\r\n\\dfrac{2}{\\left\\{ (2n+1) \\pi \\right\\}^a} \\leqq B _ n \\leqq \\dfrac{2}{\\left( 2n \\pi \\right)^a}\r\n\\]<\/li>\r\n (2)<\/strong>\u3000\u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{A _ n}{B _ n}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{A _ n}{C _ n}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(2n \\pi \\leqq x \\leqq (2n+1) \\pi\\) \u306b\u304a\u3044\u3066, \\(\\dfrac{1}{x^a}\\) \u306f\u5358\u8abf\u6e1b\u5c11\u3067\u3042\u308a, \u307e\u305f \\(\\sin x \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\n\\dfrac{\\sin x}{\\{ (2n+1) \\pi \\}^a} \\leqq \\dfrac{\\sin x}{x^a} \\leqq \\dfrac{\\sin x}{( 2n \\pi )^a} \\quad ... [1] \\ .\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\r\n\\displaystyle\\int _ {2n \\pi}^{(2n+1) \\pi} \\sin x \\, dx = [ -\\cos x ] _ {2n \\pi}^{(2n+1) \\pi} = 2 \\ .\r\n\\]\r\n\u306a\u306e\u3067, [1] \u306e\u8fba\u3005\u3092\u7a4d\u5206\u3057\u3066\r\n\\[\r\n\\dfrac{2}{\\left\\{ (2n+1) \\pi \\right\\}^a} \\leqq B _ n \\leqq \\dfrac{2}{\\left( 2n \\pi \\right)^a} \\ .\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\[\r\nA _ n = \\dfrac{1}{2} \\cdot \\pi \\cdot \\dfrac{1}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} = \\dfrac{\\pi}{2} \\cdot \\dfrac{1}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} \\quad ... [2] \\ .\r\n\\]\r\n\u3053\u308c\u3068, (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{\\pi}{4} \\cdot \\dfrac{( 2n \\pi )^a}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} & \\leqq \\dfrac{A _ n}{B _ n} \\leqq \\dfrac{\\pi}{4} \\cdot \\dfrac{\\{ (2n+1) \\pi \\}^a}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} \\\\\r\n\\text{\u2234} \\quad \\dfrac{\\pi}{4} \\cdot \\left( 1 -\\dfrac{2}{4n+1} \\right)^a & \\leqq \\dfrac{A _ n}{B _ n} \\leqq \\dfrac{\\pi}{4} \\cdot \\left( 1 +\\dfrac{2}{4n+1} \\right)^a \\quad ... [3] \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\left( 1 \\pm \\dfrac{2}{4n+1} \\right) = 1 \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(n \\rightarrow \\infty\\) \u306e\u3068\u304d\r\n\\[\r\n( [3] \\text{\u306e\u7b2c1\u8fba} ) \\rightarrow \\dfrac{\\pi}{4} , \\ ( [3] \\text{\u306e\u7b2c3\u8fba} ) \\rightarrow \\dfrac{\\pi}{4} \\ .\r\n\\]\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{A _ n}{B _ n} = \\underline{\\dfrac{\\pi}{4}} \\ .\r\n\\]\r\n (3)<\/strong><\/p>\r\n \\(\\sin^2 x \\geqq 0\\) \u306a\u306e\u3067, (1)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ {2n \\pi}^{(2n+1) \\pi} \\sin^2 x \\, dx & = \\dfrac{1}{2} \\displaystyle\\int _ {2n \\pi}^{(2n+1) \\pi} ( 1 -\\cos 2x ) \\, dx \\\\\r\n& = \\dfrac{1}{2} \\left[ x -\\dfrac{1}{2} \\sin 2x \\right] _ {2n \\pi}^{(2n+1) \\pi} = \\dfrac{\\pi}{2} \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\dfrac{\\pi}{2} \\cdot \\dfrac{1}{\\left\\{ (2n+1) \\pi \\right\\}^a} \\leqq C _ n \\leqq \\dfrac{\\pi}{2} \\cdot \\dfrac{2}{\\left( 2n \\pi \\right)^a} \\ .\r\n\\]\r\n\u3053\u308c\u3068 [2] \u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{( 2n \\pi )^a}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} & \\leqq \\dfrac{A _ n}{C _ n} \\leqq \\dfrac{\\{ (2n+1) \\pi \\}^a}{\\left\\{ \\left( 2n +\\frac{1}{2} \\right) \\pi \\right\\}^a} \\\\\r\n\\text{\u2234} \\quad \\left( 1 -\\dfrac{2}{4n+1} \\right)^a & \\leqq \\dfrac{A _ n}{C _ n} \\leqq \\left( 1 +\\dfrac{2}{4n+1} \\right)^a \\ .\r\n\\end{align}\\]\r\n (2)<\/strong> \u3068\u540c\u69d8\u306b, \u306f\u3055\u307f\u3046\u3061\u306e\u539f\u7406\u3088\u308a\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{A _ n}{C _ n} = \\underline{1} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a \\gt 0\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \\(n = 1, 2, 3, \\cdots\\) \u306b\u5bfe\u3057, \u5ea7\u6a19\u5e73\u9762\u306e \\(3\\) \u70b9 \\[ ( 2n \\pi , 0 ) , \\ \\left( \\left( 2n +\\dfrac{ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n