Linked Questions

7
votes
1answer
1k views

Limit using Poisson distribution [duplicate]

Show using the Poisson distribution that $$\lim_{n \to +\infty} e^{-n} \sum_{k=1}^{n}\frac{n^k}{k!} = \frac {1}{2}$$
13
votes
2answers
370 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( ...
11
votes
2answers
134 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 ...
3
votes
2answers
70 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) ...
6
votes
0answers
288 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
1answer
85 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.
8
votes
1answer
129 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 ...
4
votes
1answer
116 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 $x\in\mathbb{R}$, ...
2
votes
0answers
60 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
52 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
27 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
2
votes
1answer
44 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
32 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

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 ...
156
votes
81answers
13k views

Surprising identities / equations

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 ...
13
votes
4answers
452 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
286 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 ...
5
votes
2answers
147 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 ...
5
votes
1answer
248 views

A limit involves series and factorials

Evaluate : $$\lim_{n\to \infty }\frac{n!}{{{n}^{n}}}\left( \sum\limits_{k=0}^{n}{\frac{{{n}^{k}}}{k!}-\sum\limits_{k=n+1}^{\infty }{\frac{{{n}^{k}}}{k!}}} \right)$$
3
votes
4answers
184 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 ...
6
votes
2answers
189 views

High school contest question

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

How do I calculate the limit of this integral?

Using an appropriate probability distribution or otherwise show that $$\lim_{n\to\infty} \int_0^n e^{-x}{x^{n-1}\over(n-1)!}dx =0.5$$
1
vote
1answer
67 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 ...