## How to compute this series: $\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$

How to compute this series: $$\sum_{k=0}^\infty \frac{C_k}{2^{2k+1}}$$ where $$C_k$$ is the catalan number: $$C_k=\frac{1}{k+1}{2k \choose k}$$. (Further, is there any general method to treat this question with general $$C_k$$?)