\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b.\r\n\u5e73\u9762\u4e0a\u306b \\(n+2\\) \u500b\u306e\u70b9O, \\(\\text{P} {} _ 0 , \\text{P} {} _ 1 , \\cdots , \\text{P} {} _ n\\) \u304c\u3042\u308a,\r\n\u6b21\u306e \\(2\\) \u3064\u306e\u6761\u4ef6\u3092\u307f\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n
\u7dda\u5206 \\(\\text{P} {} _ {k-1} \\text{P} {} _ {k}\\) \u306e\u9577\u3055\u3092 \\(a _ k\\) \u3068\u3057, \\(s _ k = \\textstyle\\sum\\limits _ {k=1}^{n} a _ k\\) \u3068\u304a\u304f\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} s _ n\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n
\u6761\u4ef6\u3088\u308a, \\(\\triangle \\text{OP} {} _ {k-2} \\text{P} {} _ {k-1} \\sim \\triangle \\text{OP} {} _ {k} \\text{P} {} _ {k-1}\\) \u3067\u3042\u308a, \u76f8\u4f3c\u6bd4\u306f \\(1+\\dfrac{1}{n} : 1\\) \u3067\u3042\u308b.
\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na _ {k+1} = \\left( 1+\\dfrac{1}{n} \\right)\r\n\\]\r\n\u307e\u305f, \\(\\triangle \\text{OP} {} _ {0} \\text{P} {} _ {1}\\) \u306b\u3064\u3044\u3066\u4f59\u5f26\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{align}\r\na _ 1 & = \\sqrt{1^2 +\\left( 1+\\dfrac{1}{n} \\right)^2 -2 \\cdot 1 \\left( 1+\\dfrac{1}{n} \\right) \\cos \\dfrac{\\pi}{n}} \\\\\r\n& = \\dfrac{1}{n} \\sqrt{2 n (n+1) +1 -2n (n+1) \\left( 1 -2 \\sin^2 \\dfrac{\\pi}{2n} \\right)} \\\\\r\n& = \\dfrac{1}{n} \\sqrt{1 +4n(n+1) \\sin^2 \\dfrac{\\pi}{2n}}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\na _ k = \\left( 1+\\dfrac{1}{n} \\right)^{k-1} a _ 1\r\n\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\ns _ k & = a _ 1 \\textstyle\\sum\\limits _ {k=0}^{n} \\left( 1+\\dfrac{1}{n} \\right)^{k-1} \\\\\r\n& = a _ 1 \\cdot \\dfrac{\\left( 1 +\\frac{1}{n} \\right)^n -1}{1 +\\frac{1}{n} -1} \\\\\r\n& = \\left\\{ \\left( 1+\\dfrac{1}{n} \\right)^n -1 \\right\\} \\sqrt{1 +\\dfrac{4n(n+1)}{\\frac{4n^2}{\\pi^2}} \\left( \\dfrac{\\sin \\frac{\\pi}{2n}}{\\frac{\\pi}{2n}} \\right)^2} \\\\\r\n& \\rightarrow ( e-1 ) \\sqrt{1 +\\pi^2} \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} s _ n = \\underline{( e-1 ) \\sqrt{1 +\\pi^2}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \u5e73\u9762\u4e0a\u306b \\(n+2\\) \u500b\u306e\u70b9O, \\(\\text{P} {} _ 0 , \\text{P} {} _ 1 , \\cdots , \\text{P} {} _ n\\) \u304c\u3042 … \u7d9a\u304d\u3092\u8aad\u3080