3
votes
1answer
498 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 ...
8
votes
1answer
2k views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
14
votes
2answers
876 views

Calculate limit with summation index in formula [duplicate]

Possible Duplicate: Compute the limit: $\lim_{n\rightarrow\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$ I want to calculate the following: $$ \lim_{n \rightarrow \infty} \left( e^{-n} \...
11
votes
2answers
233 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 ...
2
votes
2answers
84 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) $\...
2
votes
1answer
191 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.
6
votes
0answers
351 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 ...
8
votes
1answer
156 views

How can I compute this limit? [duplicate]

I have to compute $$ \lim_{n\to\infty} \exp(-n)\left(1+n+\frac{n^2}{2}+\ldots+\frac{n^n}{n!} \right)$$ I think the value is 1, but i don't know how to proof this. Do I have to estimate the remainder ...
3
votes
1answer
89 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} ...
4
votes
1answer
142 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 $x\in\mathbb{R}$, $$\lim_{n\...
1
vote
2answers
89 views

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 ...
1
vote
0answers
85 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 \...
0
votes
2answers
39 views

Central value of the partial exponential function [duplicate]

I need help calculating the central value of the partial exponential function : $$\lim_{n \to \infty} e^{-n} \sum^n_{k=0} \frac{n^k}{k!}$$ fd
1
vote
1answer
49 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 ...
4
votes
0answers
79 views

Proving $\lim e^{-n}(1+n+n^2/2!+\cdots +n^n/n!) = 1/2$ [duplicate]

In my probability class we saw an amazing proof using the CLT (central limit theorem), I'm wondering what's a more straightforward way to do this.
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.
2
votes
1answer
66 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 ...
1
vote
0answers
49 views

Show that $\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ [duplicate]

$\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ Tried: here suppose N is poission distribution with parameter n $\lim_{n\rightarrow \infty} \sum_{i=0}^{n} P(N\...
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
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 ...
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.
0
votes
0answers
32 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 ...
212
votes
92answers
18k views

Surprising identities / equations [closed]

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 ...
14
votes
2answers
2k views

Partial sums of exponential series

What is known about $f(k)=\sum_{n=0}^{k-1} \frac{k^n}{n!}$ for large $k$? Obviously it is is a partial sum of the series for $e^k$ -- but this partial sum doesn't reach close to $e^k$ itself because ...
14
votes
4answers
565 views

Large $n$ asymptotic of $\int_0^\infty \left( 1 + x/n\right)^{n-1} \exp(-x) \, \mathrm{d} x$

While thinking of 71432, I encountered the following integral: $$ \mathcal{I}_n = \int_0^\infty \left( 1 + \frac{x}{n}\right)^{n-1} \mathrm{e}^{-x} \, \mathrm{d} x $$ Eric's answer to the linked ...
4
votes
2answers
323 views

maybe this sum have approximation $\sum_{k=0}^{n}\binom{n}{k}^3\approx\frac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty$

prove or disprove this $$\sum_{k=0}^{n}\binom{n}{k}^3\approx\dfrac{2}{\pi\sqrt{3}n}\cdot 8^n,n\to\infty?$$ this problem is from when Find this limit $$\lim_{n\to\infty}\dfrac{\displaystyle\sum_{k=...
10
votes
1answer
171 views

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, I ...
8
votes
2answers
113 views

Compute the limit $\lim_{n\to\infty}\frac{1}{\log n}\sum_{k=1}^n\left(1-\frac{1}{n}\right)^k\frac{1}{k}$

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 = \sum_{k=1}^{\...
3
votes
4answers
415 views

Expected number of tosses before you see a repeat.

Suppose we roll a fair die until some face has appeared twice. For instance, we might have a run of rolls 12545 or 636. How many rolls on average would we make? What if we roll until a face has ...
5
votes
2answers
159 views

Evaluate the limit $\lim\limits_{n\to\infty}{\frac{n!}{n^n}\bigg(\sum_{k=0}^n{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}}\bigg)}$

Evaluate the limit $$ \lim_{n\rightarrow\infty}{\frac{n!}{n^{n}}\left(\sum_{k=0}^{n}{\frac{n^{k}}{k!}}-\sum_{k=n+1}^{\infty}{\frac{n^{k}}{k!}} \right)} $$ I use $$e^{n}=1+n+\frac{n^{2}}{2!}+\...
6
votes
1answer
301 views

A limit involves series and factorials

Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$
10
votes
1answer
303 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 ...
6
votes
2answers
212 views

High school contest question

Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it? $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
6
votes
2answers
155 views

How to prove $\frac{n^n}{3n!}<\frac{e^n}{2}-\sum_{k=0}^{n-1}\frac{n^k}{k!}<\frac{n^n}{2n!}$

I met this problem: prove: $\displaystyle \frac{n^n}{3n!}<\frac{e^n}{2}-\sum_{k=0}^{n-1}\frac{n^k}{k!}<\frac{n^n}{2n!}$ I tried expand $e^n$ at $x=0$ then: $\displaystyle e^n=\sum_{k=0}^{n}\...
2
votes
3answers
90 views

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 convergence ...
1
vote
2answers
232 views

How do I calculate the limit of this integral?

Using an appropriate probability distribution or otherwise show that $$\lim_{n\to\infty} \int_0^n e^{-x}{x^{n-1}\over(n-1)!}dx =0.5$$
3
votes
1answer
165 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 \infty}\frac{1+\...
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 ...
3
votes
1answer
57 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 $X_i\...