求数列之和1/S1+1/S2+1/S3+```````+1/Sn的值我知道 Sn＝n^2+2n怎么求啊
问题描述：
求数列之和
1/S1+1/S2+1/S3+```````+1/Sn
的值
我知道 Sn＝n^2+2n
怎么求啊
答
sn=n^2+2n=n(n+2)
1/s1+1/s2+.....+1/sn
=1/(1*3)+1/(2*4)+......1/(n-1)*（n+1)+1/(n）*（n+2)
=1/2(1-1/3+1/2-1/4+......1/n-1-1/n+1+1/n-1/n+2)
=1/2(1+1/2-1/n+1-1/n+2)
=3/4-(2n+3)/2(n+2)(n+1)
这是我自己做的。你看看吧
答
Sn＝n^2+2n
1/Sn=1/(n^2+2n)=1/n(n+2)=1/2*[1/n-1/(n+2)]
1/S1+1/S2+1/S3+```````+1/Sn=1/2×[(1-1/3)+(1/2-1/4)+(1/3-1/5)+(1/4-1/6)+…+1/n-1/(n+2)]
=1/2×[1+1/2-1/(n+1)-1/(n+2)]
=3/4-(2n+3)/[2(n+1)(n+2)]