证明题.求证(1-1/3)(1-1/3^2)(1-1/3^3)×.×(1-1/3^n)>1/2

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证明题.求证(1-1/3)(1-1/3^2)(1-1/3^3)×.×(1-1/3^n)>1/2
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证明题.求证(1-1/3)(1-1/3^2)(1-1/3^3)×.×(1-1/3^n)>1/2
证明题.求证(1-1/3)(1-1/3^2)(1-1/3^3)×.×(1-1/3^n)>1/2

证明题.求证(1-1/3)(1-1/3^2)(1-1/3^3)×.×(1-1/3^n)>1/2
(1-1/3)(1-1/3^2)=1 - 1/3 - (1-1/3) * 1/3^2>1 - 1/3 - 1 * 1/3^2=1 - 1/3 - 1/3^2类似地处理n次,得 (1-1/3)(1-1/3^2)(1-1/3^3)…(1-3^n)>1 - 1/3 - 1/3^2 - 1/3^3 … - 1/3^n=2 - ( 1 + 1/3 + 1/3^2 + 1/3^3 … + 1/3^n )=2 - 1/(1 - 1/3) + 1/3^(n+1)=2 - 3/2 + 1/3^(n+1)=1/2 + 1/3^(n+1)>1/2
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证:
原式
= 1 - 1/3 - (1 - 1/3)(1/3^2) - (1 - 1/3)(1 - 1/3^2)(1/3^3) - ... - (1 - 1/3)(1 - 1/3^2)...(1/3^n)
> 1 - 1/3 - 1/3^2 - ... - 1/3^n
= 2 - (1/3^0 + 1/3^1 + 1/3^2 + ... + 1/3^n)
= 2 - [1/3^(n+1) - 1]/(1/3 - 1)
> 2 - (-1)/(1/3-1)
= 1/2