How to sum up this series? How to sum up this series :
$$2C_o + \frac{2^2}{2}C_1 +  \frac{2^3}{3}C_2 + \cdots +  \frac{2^{n+1}}{n+1}C_n$$
Any hint that will lead me to the correct solution will be highly appreciated.
EDIT: Here $C_i = ^nC_i $

Let's assume $C_i=\binom ni$. I'll give a solution that is not precalculus level. Consider first the equality
$$ (1+x)^n=C_0+xC_1+x^2C_2+\dots+x^nC_n. $$
This is the binomial theorem.
Integrate from 0 to t. On the left hand side we get $\frac{(1+t)^{n+1}-1}{n+1}$ and on the right hand side $\sum \frac1{i+1}t^{i+1}C_i$. 
Now set $t=2$, and a bit of algebra gives you the answer you want.
Pretty sure there is an elementary approach as well. 
