Linked Questions

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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 ...
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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. $$\lim_{n\rightarrow \infty} e^{-n}\cdot ... 1answer 46 views Does n power of e grow much more faster than its Maclaurin polynomial? [duplicate] I wonder how to calculate the following limit:$$ \lim_{n\rightarrow\infty}\frac{1+n+\frac{{}n^{2}}{2!}+\cdots +\frac{n^{n}}{n!}}{e^{n}} $$In the first sight, I think it should be zero, because ... 1answer 98 views Summation of exponential series [duplicate] Evaluate the limit:$$ \lim_{n \to \infty}e^{-n}\sum_{k = 0}^n \frac{n^k}{k!} $$It is not as easy as it seems and the answer is definitely not 1. Please help in solving it. 1answer 123 views Is the sequences\{S_n\} convergent? [duplicate] Let$$S_n=e^{-n}\sum_{k=0}^n\frac{n^k}{k!}$$Is the sequences\{S_n\} convergent? The following is my answer,but this is not correct. please give some hints. For all x\in\mathbb{R}, ... 2answers 148 views Weird limit \lim \limits_{n\mathop\to\infty}\frac{1}{e^n}\sum \limits_{k\mathop=0}^n\frac{n^k}{k!}  [duplicate]$$\lim \limits_{n\mathop\to\infty}\frac{1}{e^n}\sum \limits_{k\mathop=0}^n\frac{n^k}{k!} $$I thought this limit was obviously 1 at first but approximations on Mathematica tells me it's 1/2. Why ... 2answers 72 views Limits Problem : \lim_{n \to \infty}[(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})]^{\frac{1}{n}} is equal to.. [duplicate] Problem: How to find the following limit :$$\lim_{n \to \infty}[(1+\frac{1}{n})(1+\frac{2}{n})\cdots(1+\frac{n}{n})]^{\frac{1}{n}} is equal to (a) $\frac{4}{e}$ (b) $\frac{3}{e}$ (c) ...

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