\\(r\\) \u3092 \\(0 \\lt r \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9a\u6570\u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u6570\u5217 \\(\\{ a _ n \\}\\) \u3092 \\(a _ n = \\left[ \\dfrac{n}{3} \\right]\\) \u3067\u5b9a\u3081\u308b. \u305f\u3060\u3057, \u5b9f\u6570 \\(x\\) \u306b\u5bfe\u3057\u3066, \\([x]\\) \u306f \\(\\ell \\leqq x \\lt \\ell +1\\) \u3092\u307f\u305f\u3059\u6574\u6570 \\(\\ell\\) \u3092\u8868\u3059. \u3053\u306e\u3068\u304d\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\textstyle\\sum\\limits _ {k=1}^{3n} (-1)^{k-1} r^{a _ k}\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u6570\u5217 \\(\\{ b _ n \\}\\) \u3092\r\n\\[\\begin{align}\r\nn \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} \\quad & b _ n = n \\\\\r\nn \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} \\quad & b _ n = 2n\r\n\\end{align}\\]\r\n\u3067\u5b9a\u3081\u308b. \u3053\u306e\u3068\u304d\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{1}{n} \\textstyle\\sum\\limits _ {k=1}^{2n} (-1)^{k-1} r^{\\frac{b _ k}{n}}\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(n = 3m , 3m+1 , 3m+2\\) \uff08 \\(m\\) \u306f\u6574\u6570\uff09\u306b\u5bfe\u3057\u3066, \\(a _ n = m\\) \u306a\u306e\u3067,\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k = 3m}^{3m+2} (-1)^{k-1} r^{a _ k} & = (-1)^{3m-1} r^m ( 1 -1 +1 ) \\\\\r\n& = r (-r)^{m-1}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 \\(S _ m\\) \u3068\u304a\u3051\u3070, \u6c42\u3081\u308b\u6975\u9650\u5024\u306f\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^{3n} (-1)^{k-1} r^{a _ k} & = 1 -1 +\\textstyle\\sum\\limits _ {m=1}^{n-1} S _ m +(-1)^{3n-1} r^n \\\\\r\n& \\rightarrow r \\cdot \\dfrac{1}{1+r} +0 \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\\\\r\n& = \\underline{\\dfrac{r}{1+r}}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(n = 2m-1 , 2m\\) \uff08 \\(m\\) \u306f\u6574\u6570\uff09\u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\textstyle\\sum\\limits _ {k=2m-1}^{2m} (-1)^{k-1} r^{\\frac{b _ k}{n}} = r^{\\frac{2m-1}{n}} -r^{\\frac{2m}{n}}\r\n\\]\r\n\u3053\u308c\u3092 \\(T _ m\\) \u3068\u304a\u3051\u3070, \u6c42\u3081\u308b\u6975\u9650\u5024\u306f\r\n\\[\\begin{align}\r\n\\dfrac{1}{n} \\textstyle\\sum\\limits _ {k=1}^{2n} (-1)^{k-1} r^{\\frac{b _ k}{n}}\r\n& = \\dfrac{1}{n} \\textstyle\\sum\\limits _ {m=1}^{n} T _ m \\\\\r\n& = \\dfrac{1}{n} \\left( r^{\\frac{1}{n}} \\textstyle\\sum\\limits _ {m=0}^{n-1} r^{\\frac{2m}{n}} -\\textstyle\\sum\\limits _ {m=1}^{n} r^{\\frac{4m}{n}} \\right) \\\\\r\n& \\rightarrow 1 \\cdot \\displaystyle\\int _ 0^1 r^{2x} \\, dx -\\displaystyle\\int _ 0^1 r^{4x} \\, dx \\quad ( \\ n \\rightarrow \\infty \\ \\text{\u306e\u3068\u304d} \\ ) \\\\\r\n& = \\left[ \\dfrac{r^{2x}}{2 \\log r} -\\dfrac{r^{4x}}{4 \\log r} \\right] _ 0^1 \\\\\r\n& = \\underline{\\dfrac{-r^4 +2 r^2 -1}{4 \\log r}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(r\\) \u3092 \\(0 \\lt r \\lt 1\\) \u3092\u307f\u305f\u3059\u5b9a\u6570\u3068\u3059\u308b. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u6570\u5217 \\(\\{ a _ n \\}\\) \u3092 \\(a _ n = \\left[ \\dfrac{n}{3} \\right]\\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n