If the nth partial sum of a series an is sn=3-(n*2^-n) find an and sum...

...from 1 to infinity of an? so the problem as stated above is:
the nth partial sum of a series from 1 to infinity of an is Sn=3-(n*2^-n)
find an and sum from 1 to infinity of an.

so i can't figure out what i am doing wrong.
i know an= Sn-S(n-1) right?
so an=3-(n/2^n)-(3-(n-1)/(2^(n-1))
which simplifies to an=(n-2)/2^n

so i don't know where i went wrong in that. because then when i try to take the sum from 1 to infinity, i tried in two ways.
first i took the limit of Sn as n goes to infinity, which equals 3
however, when i took the sum of the an that i got from 1 to infinity i get 0.
so where did i go wrong?

thanks!