I am stuck on this problem:
Simplify the sum $$ \sum_{k=0}^n {n+k \choose k} 2^{-k}. $$
I multiplied it by $\frac{2^n}{2^n}$ to get $$ \frac{1}{2^n}\sum_{k=0}^n {n+k \choose k} 2^{n-k} = \frac{1}{2^n}\sum_{k=0}^n {n+k \choose n} 2^{n-k} $$ How should I proceed? I tried to use the formula $\sum_{i=r}^n {i \choose r} = {n+1 \choose r+1}$ but I don't know how to aply it here.
