题目

数列Cn=n(1/2)^n,求前n项和Sn.

提问时间：2020-11-23

答案

Sn=1*（1/2）+2*(1/2)^2+3*(1/2）^3+-----+(n-1)*(1/2)^(n-1)+n*(1/2)^n；
Sn/2=1*（1/2）^2+2*(1/2)^3+3*(1/2）^4+-----+(n-1)*(1/2)^n+n*(1/2)^（n+1）；
Sn-Sn/2=1*（1/2）+(2-1)*（1/2）^2+(3-2)*(1/2)^3+-----+(n-n+1)*(1/2)^n-n*(1/2)^(n+1)
= Sn/2 =（1/2）+（1/2）^2+(1/2)^3+-----+(1/2)^n-n*(1/2)^(n+1)
=(1/2)(1-(1/2)^n)/(1-(1/2))-n*(1/2)^(n+1)=1-(1/2)^n-n*(1/2)^(n+1)
=1-(n+2)*(1/2)^(n+1)
Sn=2-(n+2)/2^n