lim
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.
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