推导不等式,强人进

题目

推导不等式,强人进
1/2^k+1/(2^k+1)+…+1/(2^(k+1)-1)>2^k/(2^(k+1)-1)怎么来的?

答案

解析,
设t=2^k>1
因此,1/2^k+1/(2^k+1)+…+1/(2^(k+1)-1)
=1/t+1/(t+1)+1/(t+2)+……+1/(2t-1)
从t到(2t-1)共有t项,t>1
故,1/t+1/(t+1)+1/(t+2)+……+1/(2t-1)>t/(2t+1)
把t=2^k替换掉,
故,1/(2^k)+1/(2^k+1)+…+1/(2^(k+1)-1)>2^k/(2^(k+1)-1).