在数列an中,an=1/(n*(n+1)*(n+2)),那么Sn的极限是?

题目

在数列an中,an=1/(n*(n+1)*(n+2)),那么Sn的极限是?
最后1-2+1/2不是应该等于-1/4吗？

答案

首先分析an=1/(n*(n+1)*(n+2))=1/2{1/[n(n+1)]-1/[(n+1)(n+2)]}
所以Sn=a1+a2+...an
=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*[n(n+1)]-1/[(n+1)(n+2)]
=1/2{1/(1*2)-1/(2*3)+1/(2*3)-1/(3*4)+...+n(n+1)-1/[(n+1)(n+2)]}
=1/2*[1/2-1/(n^2+3n+2)]
所以limSn=lim{1/2*[1/2-1/(n^2+3n+2)]}=lim(1/4)=1/4