limits - Find lim$_{n to infty} sum _{ k =0}^ n frac{e^{-n}n^k}{k!}$

We need to find out the limit of,

lim$_{n \to \infty} \sum _{ k =0}^ n \frac{e^{-n}n^k}{k!}$

One can see that $\frac{e^{-n}n^k}{k!}$ is the cdf of Poisson distribution with parameter $n$.

Please give some hints on how to find out the limit.

Answer

It's a good start to try to solve it in a probabilistic way: notice that the Poisson random variable has the reproducibility property, that is, if $X_{k} \sim \text{Poisson}(1)$, $k = 1, 2, \ldots, n$ independently, then
$$S_n = \sum_{k = 1}^n X_{k} \sim \text{Poisson}(n),$$

whose distribution function $F_{S_n}$ satisfies:
$$F_{S_n}(n) = P[S_n \leq n] = \sum_{k = 0}^n e^{-n} \frac{n^k}{k!},$$
which is exactly the expression of interest. Hence this suggests linking this problem to central limit theorem.

By the classic CLT, we have
$$\frac{S_n - n}{\sqrt{n}} \Rightarrow \mathcal{N}(0, 1).$$
Hence
$$P[S_n \leq n] = P\left[\frac{S_n - n}{\sqrt{n}} \leq 0\right] \to P[Z \leq 0] = \frac{1}{2}$$
as $n \to \infty$.