2
votes
1answer
64 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
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.
207
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 ...
13
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
549 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
322 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 ...
9
votes
1answer
155 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, ...
8
votes
2answers
111 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 = ...
3
votes
4answers
392 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 ...
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
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 ...
6
votes
2answers
211 views

High school contest question

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

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