3
votes
1answer
56 views

Computing $\lim_{n\to\infty}e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ with central limit theorem

Exercise: Compute the following limit by using the central limit theorem: $$ \lim_{n\to\infty}e^{-n}\sum_{k=0}^n \frac{n^k}{k!} $$ Solution (from textbook): Let $X_1,X_2,\ldots$ i.i.d. with ...
3
votes
1answer
481 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 ...
3
votes
1answer
88 views

The limit $\lim_{n\to \infty}\frac{T_n(n)}{e^n}$ where $T_n(x)$ is the Taylor polynomial of $e^x$ [duplicate]

From working on a problem I was lead to consider the function $\frac{T_n(n)}{e^n}$ where $T_n(x)$ is the $n$'th order Taylor polynomial of $e^x$. Numerical evidence suggest that $$\lim_{n\to \infty} ...
3
votes
1answer
162 views

Limits of sequences connected with real and complex exponential

Let us denote $S_{n}(x)=1+\frac{x}{1 !}+\frac{x^{2}}{2!}+ ... + \frac{x^{n}}{n!}$. How could be calculated the limit $$L(x)=\lim_{n\to \infty}\frac{S_{n}(n x)}{e^{n x}}=\lim_{n\to ...
2
votes
1answer
64 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 ...
2
votes
1answer
175 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.
1
vote
1answer
48 views

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 ...
1
vote
1answer
79 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 ...
6
votes
0answers
345 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 ...
2
votes
0answers
73 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.
1
vote
0answers
37 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.
1
vote
0answers
82 views

Limit Challenge [duplicate]

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 ...
1
vote
0answers
40 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.
0
votes
0answers
29 views

Central limit theorem and the sequence with general term $e^{-n} ( 1+n+ \cdots + n^n/n!)$ [Proof check] [duplicate]

As an exercise I need to find the limit of the said sequence $$e^{-n} ( 1+n+ \cdots + n^n/n!)$$ using the toolkit of probability theory. Since no solution (only hints) is provided, I would appreciate ...
0
votes
0answers
38 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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