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### Exact value of an infinite series [duplicate]

The following exercise was given to me during a course of probability. I guess that this result can be used to check the Lyapunov condition of the Central Limit Theorem. Useful or not, I need to prove ...
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### Computing $\lim_{n\to\infty}e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ with central limit theorem

Exercise: Compute the following limit by using the central limit theorem: $$\lim_{n\to\infty}e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ Solution (from textbook): Let $X_1,X_2,\ldots$ i.i.d. with ...
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### Find lim$_{n \to \infty} \sum _{ k =0}^ n \frac{e^{-n}n^k}{k!}$ [duplicate]

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 ...
### Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$
$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ? Inspired by this frequently asked question, ...