\\(n\\) \u306f \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3067\u3042\u308a, \\(\\dfrac{1}{2} \\lt a _ j \\lt 1 \\quad ( j = 1, 2, \\cdots, n )\\) \u3067\u3042\u308b\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\\begin{align}\r\n( 1 -a _ 1 ) & ( 1 -a _ 2 ) \\cdots ( 1 -a _ n ) \\\\\r\n& \\gt 1 -\\left( a _ 1 +\\dfrac{a _ 2}{2} + \\cdots + \\dfrac{a _ n}{2^{n-1}} \\right)\r\n\\end{align}\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p>\r\n
\u4e0e\u3048\u3089\u308c\u305f\u4e0d\u7b49\u5f0f\u3092 [\uff0a] \u3068\u3057, [\uff0a] \u304c \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n
1*<\/strong>\u3000\\(n=2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n& ( 1-a _ 1 )( 1-a _ 2 ) -\\left\\{ 1 -\\left( a _ 1 +\\dfrac{a _ 2}{2} \\right) \\right\\} \\\\\r\n& \\qquad = a _ 1 a _ 2 -\\dfrac{a _ 2}{2} = \\left( a _ 1 -\\dfrac{1}{2} \\right) a _ 2 \\gt 0\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n=2\\) \u306e\u3068\u304d\u306f[\uff0a]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n 2*<\/strong>\u3000\\(n=k \\quad ( k \\geqq 2 )\\) \u306e\u3068\u304d[\uff0a]\u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b.\r\n\\[\\begin{align}\r\nS _ k & = ( 1-a _ 1 )( 1 -a _ 2 ) \\cdots ( 1-a _ k ) , \\\\\r\nT _ k &= 1 -\\left( a _ 1 +\\dfrac{a _ 2}{2} + \\cdots + \\dfrac{a _ k}{2^{k-1}} \\right)\r\n\\end{align}\\]\r\n\u3068\u304a\u3051\u3070\r\n\\[\r\nS _ k \\gt T _ k\r\n\\]\r\n\u3053\u308c\u3092\u5229\u7528\u3059\u308c\u3070\r\n\\[\\begin{align}\r\nS _ {k+1} -T _ {k+1} & = S _ k( 1-a _ {k+1} ) -T _ k -\\dfrac{a _ {k+1}}{2^k} \\\\\r\n& \\gt S _ k a _ {k+1} -\\dfrac{a _ {k+1}}{2^k} \\\\\r\n& = \\left( S _ k -\\dfrac{1}{2^k} \\right) a _ {k+1} \\\\\r\n& \\gt 0 \\quad \\left( \\ \\text{\u2235} \\ S _ k \\gt \\left( 1 -\\dfrac{1}{2} \\right)^k = \\dfrac{1}{2^k} \\ \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(n=k+1\\) \u306e\u3068\u304d\u3082[\uff0a]\u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n 1*<\/strong>, 2*<\/strong> \u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u306f \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3067\u3042\u308a, \\(\\dfrac{1}{2} \\lt a _ j \\lt 1 \\quad ( j = 1, 2, \\cdots, n )\\) \u3067\u3042\u308b\u3068\u304d, \u4e0d\u7b49\u5f0f \\[\\begin{ali … \u7d9a\u304d\u3092\u8aad\u3080