Linked Questions

11
votes
2answers
98 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 ...
132
votes
72answers
11k 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 ...
3
votes
2answers
65 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) ...
0
votes
0answers
27 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 ...
2
votes
0answers
58 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.
8
votes
1answer
128 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 ...
1
vote
0answers
31 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.
3
votes
4answers
148 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 ...
0
votes
0answers
28 views

How do you use the CLT to prove the following about Poisson processes? [duplicate]

I need to use the Central limit theorem to prove the following about a poisson distribution $$\lim_{n \to +\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$$ Hopefully I formatted that ...

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