Question

级数求和问题 n乘2的n次方,求和

Answer 1

Sn=1*2^1+2*2^2+3*2^3+...+n*2^n

2Sn=1*2^2+2*2^3+3*2^4+...+n*2^(n+1)

上式减下式,得：-Sn=2+(2^2+2^3+...+2^n)-n*2^(n+1)

-Sn=2(1-2^n)/(1-2)-2n*2^n

Sn=2-2^(n+1)+n*2^(n+1)

=2+(n-1)*2^(n+1)