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### 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 ...
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### Summation of exponential series [duplicate]

Evaluate the limit: $$\lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!}$$ It is not as easy as it seems and the answer is definitely not 1. Please help in solving it.
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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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### Asymptotics of $\sum_{i=0}^{n} \frac{i n^i}{i!}$.

How can you calculate the asymptotics of $\sum_{i=0}^{n} \frac{i n^i}{i!}$ ? This sum looks similar to the power series for $e^n$. I have also seen similar problems solved using Poisson distributions ...
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### Finding $\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ if it exists [duplicate]

Does there exist the following limitation? If the answer is yes, could you show me how to find that? $$\lim_{n\to\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$$ In the following, I'm going to write what ...
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### Proving $\lim e^{-n}(1+n+n^2/2!+\cdots +n^n/n!) = 1/2$ [duplicate]

In my probability class we saw an amazing proof using the CLT (central limit theorem), I'm wondering what's a more straightforward way to do this.
### $\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!}$ [duplicate]
How can be evaluated this limit: $$\lim_{n\to\infty} e^{-n}\sum_{k=1}^n \frac{n^k}{k!} .$$ Thank you.