证明:
(1)当n=2时,
左边=1+1/2+1/3+1/4=25/12
右边=(2+2)/2=2=24/12
所以左边>右边成立,即n=2时命题成立.
(2)假设当n=k(k>=2时)命题成立,
即1+1/2+1/3+...+1/2^k>(k+2)/2
则当n=k+1时,
左边=1+1/2+1/3+...+1/2^k+1/(2^k+1)+...+1/2^(k+1)
>(k+2)/2+1/2^(k+1)+1/2^(k+1)+...+1/2^(k+1)
=(k+2)/2+2^k/2^(k+1)
=(k+2)/2+1/2
=(k+1+2)/2
即n=k+1时也成立.
由(1)(2)可得原命题成立.