<A HREF="https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas">Expansion</A> of the binomial ${n\choose k}$ around the maximum $k=n/2$ gives a narrowly peaked Gaussian for large $n$, 
$$\binom{n}k=\frac{2^n}{\sqrt{n\pi/2}} \exp\left(-\frac{(k-n/2)^2}{n/2}\right) \left(1+{\cal O}(n^{-1/2})\right),$$
from which 
$$\lim_{n\rightarrow\infty} \frac{\sum_{k=0}^n k^\alpha {n\choose k}}{n^\alpha 2^{n}}=2^{-\alpha}.$$