I am interested in the limit $\frac{\sum_{k=0}^n \sqrt{k}\cdot\binom{n}{k}}{\sqrt{n}\cdot2^n}$ as $n$ goes to infinity. Any reference or argument?

In general what do we know about the asymptotic behavior of $\sum_{k=0}^n k^\alpha\cdot\binom{n}{k}$, where $\alpha$ is a positive real (not necessarily integer)?