A little work shows that the nth term
a<n> = 3/(2^(n+1)-1)
Now, do you want the limit of the terms, or the limit of the sums?
a<?> = 0
S<?> = 3(sum 1/(2^(n+1)-1)
I thought that would be relatively simple, but you can see below that it is not:
http://www.wolframalpha.com/input/?i=sum+1%2F(2%5Ek-1)