题目

lim(n->∞){1/2*4+1/3*5+...+1/(n+1)(n+3)=?

提问时间：2021-03-31

答案

1/2*4=1/2(1/2-1/4)
1/3*5=1/2(1/3-1/5)
1/4*6=1/2(1/4-1/6)
1/(n+1)(n+3)=1/2[1/(n+1)-1/(n+3)]
lim(n->∞){1/2*4+1/3*5+...+1/(n+1)(n+3)=
lim(n->∞)1/2{1/2-1/4+1/3-1/5+1/4-1/6+ +1/(n+1)-1/(n+3)]
=lim(n->∞)1/2[1/2+1/3-1/(n+2)-1/(n+3)]=5/12