急!求正整数的最大值,使不等式(1/n+1)+(1/n+2)+...+(1/3n+1)>a-7,对一切正整数n都成立.

问题描述:

急!求正整数的最大值,使不等式(1/n+1)+(1/n+2)+...+(1/3n+1)>a-7,对一切正整数n都成立.
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1个回答 分类:数学 2014-11-29

问题解答:

我来补答
设f(n) = 1/(n+1) + 1/(n+2) + ... + 1/(3n+1)
则f(n+1) = 1/(n+2) + 1/(n+3) + ... + 1/[3(n+1)+1]
= 1/(n+2) + 1/(n+3) + ... + 1/(3n+4)
则f(n)-f(n+1) = 1/(n+1) - [1/(3n+2) + 1/(3n+3) + 1/(3n+4)]
= 1/(n+1) - [1/(3n+3)+(3n+2+3n+4)/((3n+2)(3n+4))]
= 1/(n+1) - [1/(3n+3) + (6n+6)/((3n+2)(3n+4))]
【(3n+2)(3n+4)=9n^2+18n+81 /((3n+3)(3n+3))……应用到下式】
f(n)-f(n+1)< 1/(n+1) - [1/(3n+3) + (6n+6)/((3n+3)(3n+3))]
= 1/(n+1) - [1/(3n+3) + 2/(3n+3)]
= 1/(n+1) - 3/(3n+3)
= 0
因为f(n)-f(n+1)a-7对所有自然数n成立
所以只要13/12>a-7,
解得a
 
 
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