Question

已知数列1*1/2,2*1/4,3*1/8,4*1/16,···,n*1/2^n,···,求sn

Answer 1

设第n项为an

a1=1×1/2^1

a2=1×2/2^2

a3=1×3/2^3

…………

an=n/2^n

Sn=a1+a2+...+an

=1/2^1+2/2^2+3/2^3+...+n/2^n

Sn/2=1/2^2+2/2^3+...+(n-1)/2^n+n/2^(n+1)

Sn-Sn/2=Sn/2=1/2^1+1/2^2+1/2^3+...+1/2^n-n/2^(n+1)

=(1/2)[1-(1/2)^n]/(1-1/2)-n/2^(n+1)

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

Sn=2-2/2^n-n/2^n

=2-(n+2)/2^n