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

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

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