# Linked Questions

1answer
85 views

### 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 ...
90answers
15k views

### Surprising identities / equations

What are some surprising equations / identities that you have seen, which you would not have expected? This could be complex numbers, trigonometric identities, combinatorial results, algebraic ...
0answers
26 views

### Limit of a series (Gamma distribution) [duplicate]

How does one calculate $$\lim_{n\to\infty}e^{-n}\sum_{k=0}^{n-1}\frac{n^k}{k!}?$$ Numerically it is somewhat close to $\frac12$. But to prove that I am going around a circle! Thanks for any help.
1answer
143 views

2answers
102 views

1answer
53 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 ...
1answer
111 views

### 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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