Prove $\binom{n}{0} + \frac{1}{2} \binom{n}{1} + \frac{1}{3} \binom{n}{2} + ... + \frac{1}{n+1} \binom{n}{n} = \frac{2^{n + 1} - 1}{n + 1}$. I have to prove the identity:

$$\binom{n}{0} + \dfrac{1}{2} \binom{n}{1} + \dfrac{1}{3} \binom{n}{2}
 + ... + \dfrac{1}{n+1} \binom{n}{n} = \dfrac{2^{n + 1} - 1}{n + 1}$$

First thing I tried was to think about a combinatorial proof, but I couldn't interpret the identity in any way that makes sense. So, how should I prove this?

Hint: Integrate the binomial theorem 
$$(1+x)^n = \sum_{k=0}^n\binom nk x^k$$
from $x=0$ to $x=1$.
