简单计算一下即可，答案如图所示
let
S = 1.(1/2)^0+ 2.(1/2)^1+....+n.(1/2)^(n-1) (1)
(1/2)S = 1.(1/2)^1+ 2.(1/2)^2+....+n.(1/2)^n (2)
(1)-(2)
(1/2)S = (1+1/2+1/2^2+...+1/2^(n-1)) - n.(1/2)^n
= 2( 1- (1/2)^n ) - n.(1/2)^n
S =4( 1- (1/2)^n ) - 2n.(1/2)^n
an = (2n-1)/2^n
= n.(1/2)^(n-1) - 1/2^n
Sn = a1+a2+...+an
= S - (1- (1/2)^n )
= 3( 1- (1/2)^n ) - 2n.(1/2)^n
=3 -(2n+3).(1/2)^n
lim(n->∞) Sn =3
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