\u6570\u5217 \\(\\{ a _ n \\} \\ ( a _ n \\gt 0 )\\) \u3092\u6b21\u306e\u898f\u5247\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b\uff1a\r\n\\[\r\na _ 1 = 1 \\ : \\ \\displaystyle\\int _ {a _ n}^{a _ {n+1}} \\dfrac{dx}{\\sqrt[3]{x}} = 1 \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u66f2\u7dda \\(y= \\dfrac{1}{\\sqrt[3]{x}}\\) \u3068, \\(x\\) \u8ef8\u304a\u3088\u3073 \\(2\\) \u76f4\u7dda \\(x = a _ n\\) , \\(x = a _ {n+1}\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092 \\(x\\) \u8ef8\u306e\u5468\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u305f\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ x\\) \u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} \\sqrt{n} V _ x\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n
\\[\\begin{align}\r\n\\displaystyle\\int _ {a _ n}^{a _ {n+1}} \\dfrac{dx}{\\sqrt[3]{x}} & = \\left[ \\dfrac{3}{2} x^{\\frac{2}{3}} \\right] _ {a _ n}^{a _ {n+1}} \\\\\r\n& = \\dfrac{3}{2} \\left( {a _ {n+1}}^{\\frac{2}{3}} -{a _ n}^{\\frac{2}{3}} \\right) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{3}{2} \\left( {a _ {n+1}}^{\\frac{2}{3}} -{a _ n}^{\\frac{2}{3}} \\right) & = 1 \\\\\r\n\\text{\u2234} \\quad {a _ {n+1}}^{\\frac{2}{3}} & = {a _ n}^{\\frac{2}{3}} +\\dfrac{2}{3} \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6570\u5217 \\(\\left\\{ {a _ n}^{\\frac{2}{3}} \\right\\}\\) \u306f\u521d\u9805 \\({a _ 1}^{\\frac{2}{3}} = 1\\) , \u516c\u5dee \\(\\dfrac{2}{3}\\) \u306e\u7b49\u5dee\u6570\u5217\u306a\u306e\u3067\r\n\\[\r\n{a _ n}^{\\frac{2}{3}} = 1 +\\dfrac{2}{3} (n-1) = \\dfrac{2n-1}{3} \\quad ... [1] \\ .\r\n\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nV _ x & = \\pi \\displaystyle\\int _ {a _ n}^{a _ {n+1}} \\left( \\dfrac{1}{\\sqrt[3]{x}} \\right)^2 \\, dx \\\\\r\n& = \\pi \\left[ 3 x^{\\frac{1}{3}} \\right] _ {a _ n}^{a _ {n+1}} \\\\\r\n& = 3 \\pi \\left( {a _ {n+1}}^{\\frac{1}{3}} -{a _ n}^{\\frac{1}{3}} \\right) \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3068 [1] \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\sqrt{n} V _ x & = 3 \\pi \\sqrt{n} \\left( \\sqrt{\\dfrac{2n+1}{3}} -\\sqrt{\\dfrac{2n-1}{3}} \\right) \\\\\r\n& = \\sqrt{3n} \\pi \\cdot \\dfrac{2}{\\sqrt{2n+1} +\\sqrt{2n-1}} \\\\\r\n& = \\dfrac{2 \\sqrt{3} \\pi}{\\sqrt{2 +\\frac{1}{n}} +\\sqrt{2 -\\frac{1}{n}}} \\\\\r\n& \\rightarrow \\dfrac{2 \\sqrt{3} \\pi}{2 \\sqrt{2}} \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} ) \\\\\r\n& = \\dfrac{\\sqrt{6}}{2} \\pi \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\sqrt{n} V _ x = \\underline{\\dfrac{\\sqrt{6}}{2} \\pi} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6570\u5217 \\(\\{ a _ n \\} \\ ( a _ n \\gt 0 )\\) \u3092\u6b21\u306e\u898f\u5247\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b\uff1a \\[ a _ 1 = 1 \\ : \\ \\displaystyle\\int _ {a _ n}^{a _ {n+1}} … \u7d9a\u304d\u3092\u8aad\u3080