题目

分子是1分母有三项相乘的分式如何裂项!

提问时间：2021-03-19

答案

不知道你要问的是否是1/[n(n+1)(n+2)]如何裂项?比如让求
Sn=1/(1*2*3)+1/(2*3*4)+……+1/[n(n+1)(n+2)]
因1/[n(n+1)(n+2)]=1/2*2/[n(n+1)(n+2)]
=1/2*[(n+2)-n]/[n(n+1)(n+2)]
=1/2*{1/[n(n+1)]-1/[(n+1)(n+2)]}
故
Sn=1/(1*2*3)+1/(2*3*4)+……+1/[n(n+1)(n+2)]
=1/2*[1/(1*2)-1/(2*3)]+1/2*[1/(2*3)-1/(3*4)]+……+1/2*{1/[n(n+1)]-1/[(n+1)(n+2)]}
=1/2*{1/(1*2)-1/[(n+1)(n+2)]}
=1/2*[(n+1)(n+2)-2]/[2(n+1)(n+2)]
=n(n+3)/[4(n+1)(n+2)]