若an=1+2+3…+n,数列{1/an}的前n项和为Sn,则S100=

题目

若an=1+2+3…+n,数列{1/an}的前n项和为Sn,则S100=

答案

an=n(n+1)/2
1/an=2/[n(n+1)]=2*[(1/n)-1/(n+1)]
Sn=a1+2+……+an
=2*[(1/1)-1/(1+1)]+2*[(1/2)-1/(2+1)]+……+2*[(1/n)-1/(n+1)]
=2*[(1/1)-1/(n+1)]
所以S100=2*[(1/1)-1/(100+1)]
=200/101