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### Calculate the limit: $\lim_{n\to\infty} \frac{\gamma(n,n)}{\Gamma(n)}$

$$\lim_{n\to\infty} \frac{\gamma(n,n)}{\Gamma(n)}$$ where $$\gamma(s,x)=\int_0^x t^{s-1} e^{-t}$$ This limit is part of my attempt to measure the divergence rates of $\Gamma(x)$ and the ...
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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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### 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 ...
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### Find lim$_{n \to \infty} \sum _{ k =0}^ n \frac{e^{-n}n^k}{k!}$ [duplicate]

We need to find out the limit of, lim$_{n \to \infty} \sum _{ k =0}^ n \frac{e^{-n}n^k}{k!}$ One can see that $\frac{e^{-n}n^k}{k!}$ is the cdf of Poisson distribution with parameter $n$. Please ...
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### Closed Form for $~\lim\limits_{n\to\infty}~\sqrt n\cdot(-e)^{-n}\cdot\sum\limits_{k=0}^n\frac{(-n)^k}{k!}$

$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ? Inspired by this frequently asked question, ...
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Similar to this problem, how can one compute the following limit: $$\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}\quad ?$$ Note that $$\log x = ... 1answer 289 views ### Probability and Laplace/Fourier transforms to solve limits/integrals from calculus. I've seen in some answers in Brilliant.org to some very complicated limits and integrals that uses probabilistic arguments (Let X be a random variable from [0,1]... some examples are in those ... 0answers 82 views ### Limit Challenge [duplicate] I have tried very hard to manipulate this limit, but can't seem to find what the answer is. Can someone give me an explanation to what the limit would be.$$\lim_{n\rightarrow \infty} e^{-n}\cdot ...
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 ...
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.