calculus - Limit of an expression

$$\lim\limits_{n\to\infty}\frac{1}{e^n\sqrt{n}}\sum\limits_{k=0}^{\infty}\frac{n^k}{k!}|n-k|=\sqrt{2/\pi}$$
Is this limit true? I should show limit is true. It is allowed to use computer programs to find this limit.
Thanks for your helps...

Answer

This question has a nice probabilistic interpretation. Given that $X$ is a Poisson distribution with parameter $\lambda=n$, we are essentially computing the expected value of the absolute difference between $X$ and its mean $n$. The central limit theorem gives that $Y\sim N(n,n)$ (a normal distribution with mean and variance equal to $n$) is an excellent approximation of our distribution for large values of $n$, hence:

$$\begin{eqnarray*}\frac{1}{e^n \sqrt{n}}\sum_{k=0}^{+\infty}\frac{n^k}{k!}|n-k|&\approx&\frac{1}{\sqrt{n}}\cdot\frac{1}{\sqrt{2\pi n}}\int_{-\infty}^{+\infty}|x-n|\exp\left(-\frac{(x-n)^2}{2n}\right)\,dx\\&=&\frac{2}{n\sqrt{2\pi}}\int_{0}^{+\infty}x\exp\left(-\frac{x^2}{2n}\right)\,dx\\&=&\color{red}{\sqrt{\frac{2}{\pi}}},\end{eqnarray*}$$
so the limit is not zero.