求数列1*2分之2,2*3分之2,3*4分之2,4*5分之2,...的前n项和sn
问题描述：
求数列1*2分之2,2*3分之2,3*4分之2,4*5分之2,...的前n项和sn
答
Sn=2/(1×2)+2/(2×3)+2/(3×4)+2/(4×5)+...+2/[n(n+1)]
=2[1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+...+1/n(n+1)]
=2[1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+...+1/n-1/(n+1)]
=2[1-1/(n+1)]
=2n/(n+1)