# Linked Questions

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### Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
1answer
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### How to compute $\lim_{n\rightarrow\infty}e^{-n}\left(1+n+\frac{n^2}{2!}\cdots+\frac{n^n}{n!}\right)$ [duplicate]

There is a probabilistic method to solve it. But I am not familiar with probability. I am trying to compute it by analytic method, such as using L Hospital's rule or Stolz formula, but they are not ...
2answers
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0answers
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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. $$\lim_{n\rightarrow \infty} e^{-n}\cdot ... 0answers 69 views ###  \lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!}  [duplicate] How can be evaluated this limit:$$ \lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} .$$Thank you. 2answers 31 views ### Central value of the partial exponential function [duplicate] I need help calculating the central value of the partial exponential function :$$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$fd 1answer 55 views ### Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate] I wonder how to calculate the following limit:$$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$In the first sight, I think it should be zero, because ... 0answers 36 views ### Compare e^n and its first n terms sum [duplicate] Compute the limit as n approaches infinity.$$ \frac{\sum_{0\le i\le n} \frac{n^i}{i!}}{e^n}  It is somehow between 0 and 1.

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