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### Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
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### $\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!}$ [duplicate]

How can be evaluated this limit: $$\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} .$$ Thank you.
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I have tried very hard to manipulate this limit, but can't seem to find what the answer is. Can someone give me an explanation to what the limit would be. $$\lim_{n\rightarrow \infty} e^{-n}\cdot ... 1answer 48 views ### Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate] I wonder how to calculate the following limit:$$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$In the first sight, I think it should be zero, because ... 2answers 28 views ### Central value of the partial exponential function [duplicate] I need help calculating the central value of the partial exponential function :$$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$fd 0answers 33 views ### Compare e^n and its first n terms sum [duplicate] Compute the limit as n approaches infinity.$$ \frac{\sum_{0\le i\le n} \frac{n^i}{i!}}{e^n} $$It is somehow between 0 and 1. 0answers 29 views ### Limit of Series with Variable Lower Bound [duplicate] I'm trying to compute the following limit of a series:$$\lim_{n\to\infty} \sum_{k = n+1}^{\infty}\frac{e^{-n}n^{k}}{k!} Factoring $e^{-n}$ out of the sum, applying the definition of $e^{x}$ as a ...

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