$X_{n}$ has binomial distribution with $p=q=\frac{1}{2}$ and $n$ trials. How does one show $\lim_{n \to \infty} E(|2X_{n}-n|) = \sqrt{\frac{2n}{\pi}}$? (This is the expected value of the absolute displacement of a symmetric random walk). I have tried using the de Moivre-Laplace theorem but can't figure out how exactly to make the argument.
Edit: trying to do this only using deMoivre-Laplace or the Stirling approximation.