Math. Induction?

Prove by M. I. that
1－1/2+1/3－1/4+ ......+1/(2n+1)－1/(2n + 2)=1/(n + 2)+1/(n + 3)+ ..... +1/(2n + 2)
Sol
當n=1時
左=1/1-1/2+1/3-1/4=(12-6+4-3)/12=7/12
右=1/3+1/4=7/12
左＝右
n=1時為真……………………….
設 n=k時為真即
1－1/2+1/3－1/4+ ......+1/(2k+1)－1/(2k + 2)=1/(k + 2)+1/(k + 3)+ ..... +1/(2k + 2)
[1－1/2+1/3－1/4+ ......+1/(2k+1)－1/(2k + 2)]-1/(2k+2+2)
=[1/(k + 2)+1/(k + 3)+ ..... +1/(2k + 2)]-1/(2k+4)
=[1/(k + 3)+ ..... +1/(2k + 2)]+1/(k+2)-1/(2k+4)
=[1/(k + 3)+ ..... +1/(2k + 2)]+2/(2k+4)-1/(2k+4)
=[1/(k + 3)+ ..... +1/(2k + 2)]+1/(2k+4)
=1/(k + 3)+ ..... +1/(2k + 2)+1/(2k+4)
=1/[(k+1)+2)+ ..... +1/[2(k+1)+1/[2(k+1)+2]
So
n=k+1時為真