当n=2时,左边为1/2^2,右边为1/2 左边<右边
假设n=k成立,即有1/2^2+1/3^3+...+1/k^k<(k-1)/k
当n=k+1,1/2^2+1/3^3+...+1/k^k+1/(k+1)^(k+1)<(k-1)/k+1/(k+1)^(k+1)<(k-1)/k+1/(k+1)^2<(k-1)/k+1/(k+1)k=k/(k+1),即对k+1也成立
由归纳法可知,对任意大于1的n都成立