\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057 \\(t\\) \u306f \\(0 \\lt t \\lt \\pi\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u6b21\u306e\u7b49\u5f0f\u3092\u8a3c\u660e\u305b\u3088.\r\n\\[\r\n\\left( \\cos \\dfrac{t}{2} \\right) \\left( \\cos \\dfrac{t}{4} \\right) \\left( \\cos \\dfrac{t}{8} \\right) = \\dfrac{\\sin t}{8 \\sin \\dfrac{t}{8}}\r\n\\]<\/li>\r\n (2)<\/strong>\u3000\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u6570\u5217 \\(\\{ a _ n \\}\\) \u306e\u6975\u9650\u5024 \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n\\) \u3092 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.\r\n\\[\r\na _ 1 = \\cos \\dfrac{t}{2} , \\ a _ n = a _ {n-1} \\left( \\cos \\dfrac{t}{2^n} \\right) \\quad ( n =2, 3, \\cdots )\r\n\\]<\/li>\r\n (3)<\/strong>\u3000\u6570\u5217 \\(\\{ b _ n \\} , \\{ c _ n \\}\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3059\u308b.\r\n\\[\\begin{align}\r\nb _ 1 & = \\sqrt{\\dfrac{1}{2}} , \\ b _ n = \\sqrt{\\dfrac{1 +b _ {n-1}}{2}} \\quad ( n = 2, 3, \\cdots ) \\\\\r\nc _ 1 & = \\sqrt{\\dfrac{1}{2}} , \\ c _ n = c _ {n-1} b _ n \\quad ( n = 2, 3, \\cdots )\r\n\\end{align}\\]\r\n\u3053\u306e\u3068\u304d \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} c _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(2\\) \u500d\u89d2\u306e\u516c\u5f0f\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n8 \\left( \\cos \\dfrac{t}{2} \\right) & \\left( \\cos \\dfrac{t}{4} \\right) \\left( \\cos \\dfrac{t}{8} \\right) \\left( \\sin \\dfrac{t}{8} \\right) \\\\\r\n& = 4 \\left( \\cos \\dfrac{t}{2} \\right) \\left( \\cos \\dfrac{t}{4} \\right) \\left( \\sin \\dfrac{t}{4} \\right) \\\\\r\n& = 2 \\left( \\cos \\dfrac{t}{2} \\right) \\left( \\sin \\dfrac{t}{2} \\right) \\\\\r\n& = \\sin t\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n (2)<\/strong><\/p>\r\n \u6761\u4ef6\u3088\u308a\r\n\\[\r\na _ n =\\left( \\cos \\dfrac{t}{2^n} \\right) \\left( \\cos \\dfrac{t}{2^{n-1}} \\right) \\cdots \\left( \\cos \\dfrac{t}{2} \\right)\n\\]\r\n(1)<\/strong> \u3068\u540c\u69d8\u306b\u3059\u308c\u3070,\r\n\\[\r\na _ n =\\dfrac{\\sin t}{2^n \\sin \\dfrac{t}{2^n}}\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n & = \\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{\\sin t}{t \\left( \\dfrac{2^n}{t} \\sin \\dfrac{t}{2^n}\\right)} \\\\\r\n& = \\dfrac{\\sin t}{t \\cdot 1} \\quad \\left( \\ \\text{\u2235} \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d,} \\ \\dfrac{t}{2^n} \\rightarrow 0 \\ \\right) \\\\\r\n& = \\underline{\\dfrac{\\sin t}{t}}\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \u307e\u305a\r\n\\[\r\nb _ n = \\cos \\dfrac{\\pi}{2^{n+1}} \\quad ... [\\text{A}]\n\\]\r\n\u3068\u306a\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d\r\n\\[\r\nb _ 1 = \\sqrt{\\dfrac{1}{2}} = \\cos \\dfrac{\\pi}{4}\n\\]\r\n\u306a\u306e\u3067, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(n = k-1 \\ ( k \\geqq 2 )\\) \u306e\u3068\u304d \u3088\u3063\u3066, \u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [A] \u304c\u6210\u7acb\u3059\u308b.\r\n\u3053\u3053\u3067\r\n\\[\r\nc _ 1 = \\sqrt{\\dfrac{1}{2}} = \\cos \\dfrac{\\pi}{4}\n\\]\r\n\u3067\u3042\u308a\r\n\\[\r\nc _ n = c _ {n-1} \\left( \\cos \\dfrac{\\pi}{2^{n+1}} \\right)\n\\]\r\n\u306a\u306e\u3067, \u6570\u5217 \\(\\{ c _ n \\}\\) \u306f, \\(t = \\dfrac{\\pi}{2}\\) \u3068\u304a\u3044\u305f\u6570\u5217 \\(\\{ a _ n \\}\\) \u3067\u3042\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n
\r\n[A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\r\na _ {k} = \\sqrt{\\dfrac{1 +\\cos \\dfrac{\\pi}{2^k}}{2}} = \\cos \\dfrac{\\pi}{2^{k+1}}\n\\]\r\n\u306a\u306e\u3067, \\(n = k\\) \u306e\u3068\u304d\u3082, [A] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n
\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} c _ n = \\dfrac{\\sin \\dfrac{\\pi}{2}}{\\dfrac{\\pi}{2}} = \\underline{\\dfrac{2}{\\pi}}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057 \\(t\\) \u306f \\(0 \\lt t \\lt \\pi\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b. (1)\u3000\u6b21\u306e\u7b49\u5f0f\u3092\u8a3c\u660e\u305b\u3088. \\[ \\left( \\cos \\dfrac{t}{2} \\right) \\lef … \u7d9a\u304d\u3092\u8aad\u3080