1-1/2+1/3-1/4+.+1/（2n-1)-1/2n=1/(n+1)+1/(n+2）+1/（3+n).+1/2n

1-1/2+1/3-1/4+.+1/（2n-1)-1/2n=1/(n+1)+1/(n+2）+1/（3+n).+1/2n
1-1/2+1/3-1/4+.+1/（2n-1)-1/2n=1/(n+1)+1/(n+2）+1/（3+n).+1/2n

1-1/2+1/3-1/4+.+1/（2n-1)-1/2n=1/(n+1)+1/(n+2）+1/（3+n).+1/2n
数学归纳法
n=1时成立 1-1/2=1/(1+1)
n=k时若成立
则n=k+1时
左边=1/(k+1)+...+1/2k+1/(2k+1)-1/2(k+1)
=1/(k+2)+...+1/(2k+1)+1/2(k+1)
因此n=k+1时也成立
因此原式成立