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 $

REMARK $\ $ The various approaches are all equivalent. Namely, suppose that we desire to prove without calculus the identity arising from integrating the binomial formula, viz.  $$\rm (1 + x)^{n+1}\ =\ 1 + \sum_{k=1}^{n+1}\: \frac{n+1}{k+1} {n\choose k}\ x^{k+1}$$
Comparing coefficients reduces it to the identity
$$\rm \quad\quad\  {n+1 \choose k+1}\ =\ \frac{n+1}{k+1} {n\choose k} $$
which is precisely the identity employed in Moron's "calculus free" approach.
