证明:当n=2时,左边=1-1/2=1/2,右边=2(1/(2+2))=1/2,左边=右边,成立假设当n=2k时,成立,即1-1/2+1/3.....+1/(2k-1)-1/2k=2[1/(2k+2)+(1/2k+4)+.......+1/(4k)]则当n=2k+2时,左边=1-1/2+1/3.....+1/(2k-1)-1/2k+1/(2k+1)-1/(2k+2)因为1-1/2+1/3.....+1/(2k-1)-1/2k=2[1/(2k+2)+1/(2k+4)+.......+1/(4k)]所以左边=2[1/(2k+2)+1/(2k+4)+...+1/(4k)]+1/(2k+1)-1/(2k+2)=2[1/(2k+2)+1/(2k+4)+...+1/(4k)]+2/(4k+2)-2/(4k+4)=2[2/(4k+4)+1/(2k+4)+...+1/(4k)+1/(4k+2)-1/(4k+4)]=2[1/(2k+4)+1/(4k+6)+...+1/(4k)+1/(4k+2)+1/(4k+4)]=右边所以成立证毕