分式求和问题1/(2^k+1)+1/(2^k+2)+…+1/2^(k+1)

题目

分式求和问题1/(2^k+1)+1/(2^k+2)+…+1/2^(k+1)
为什么1/(2^k+1)+1/(2^k+2)+…+1/2^(k+1)
=1/(2^k+1)+1/(2^k+2)+…+1/[2^k+2^(k-1)]+1/[(2^k+2^(k-1)+1]+1/[(2^k+2^(k-1)+2]+…+1/2^(k+1)
≥2^(k-1)/[2^k+2^(k-1)]+2^(k-1)/2^(k+1)=7/12
如何由第二步到第三步的?

答案

第二步中,前半部分分母全部变为2^k+2^(k-1),后半部分分母全部变为2^(k+1),然后分别相加.
分母增大,则结果变小