# Linked Questions

1answer
105 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.
1answer
70 views

### 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 ...
0answers
306 views

### 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 ...
0answers
65 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.
0answers
24 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.
0answers
59 views

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. $$\lim_{n\rightarrow \infty} e^{-n}\cdot ... 0answers 33 views ### Compare e^n and its first n terms sum [duplicate] Compute the limit as n approaches infinity.$$ \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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