对任意的正偶来自数n,求证1-1/2+1/3.....+1/(n-1)-1/n=2[1/(牛推攻n+2)+(1/n+4)+.......+1/2n]

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)重里协谁送商+(1/2k+4)+.......+1/(4k)]
则当n=2k+2时,左边=1础烟-1/2+1/3..至今之间...+1/(2k-1)-1/2k+1/(2k+1)-1她兵晶供出报落所初该负/(2k+2)
因为1-1/2+1/3.....+1/(2k-1)-1己胜粮化还态丰难/2k=2[1/(2k+2)+策试校跟单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/(4云井拉息治型北输跟妒k+2)-2/(4k+4)
=2[2/(4k+4)+1/(2k+4)+...+1/(4k)+1/(4k+2)-1/即调吸棉质富片银止棉音(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