Question

数列1+1/2,2+3/4,3+7/8……+n+（（2^n）-1）/2^n）求和

Answer 1

设第n项为an

an=n+ (2ⁿ-1)/2ⁿ=n+1 -1/2ⁿ

Sn=a1+a2+...+an

=(1+2+...+n)+n -(1/2+1/2²+...+1/2ⁿ)

=n(n+1)/2 +n -(1/2)(1-1/2ⁿ)/(1-1/2)

=n(n+3)/2 + 1/2ⁿ -1