Tagged Questions
4
votes
3answers
59 views
Calculating limit involving factorials.
I want to show that $\lim\limits_{k\to\infty} \frac{\pi^kk!}{(2k+1)!} = 0$. I've been trying to use the squeeze theorem, but am having a hard time finding some expression $P$ involving $k$ that is ...
18
votes
4answers
357 views
Limit of series involving ratio of two factorials
$$
\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}
$$
The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
0
votes
1answer
70 views
Simplifying a factorial containing only variables
I basically know how Im supposed to do this but I cant think of how to write it out on paper so someone else can follow what I did
I need to find the limit of:
$$\displaystyle\lim_{n \to \infty} ...
5
votes
1answer
120 views
A limit involves series and factorials
Evaluate :
$$\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)$$
2
votes
3answers
47 views
Convergence of Sequence with factorial
I want to show that
$$
a_n = \frac{3^n}{n!}
$$
converges to zero. I tried Stirlings formulae, by it the fraction becomes
$$
\frac{3^n}{\sqrt{2\pi n} (n^n/e^n)}
$$
which equals
$$
...
4
votes
5answers
352 views
How can I calculate the limit of exponential divided by factorial?
I suspect this limit is 0, but how can I prove it?
$$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$
1
vote
3answers
661 views
Find the limit of exponent/factorial sequence [duplicate]
Possible Duplicate:
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I don't know how to even stoke it...
...
1
vote
4answers
216 views
Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I'm looking for a way to find this limit:
$\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I think I have found that it diverges, by plugging numbers into the formula and "sandwich" the result. However I ...
4
votes
3answers
873 views
Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$
I need to check if
$$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
5
votes
3answers
164 views
Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)
I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate:
...
4
votes
2answers
140 views
Upper bound for the series $\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$
I want to show that the series
$$\sum_{n\geq 1}\frac{1}{(n+1)^{a+1}}\sum_{k=0}^n b^k\left(\frac{(n-k)!}{n!}\right)^a$$
converges for $a,b>0$. I have tried this so much that the smallest hint will ...
2
votes
1answer
34 views
Limit of $\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}$ when $n\rightarrow\infty$
I want to prove that $$\lim_{n\rightarrow\infty}\frac{a^{n+1}(n+1)!^b}{\sum_{k=0}^n a^kk!^b}<\infty$$ for $a,b>0$.
This is the last step of a bigger problem. I believe it would suffice to use ...
11
votes
1answer
394 views
Factorial canceling on expansion of binomial coefficients on Concrete Mathematics
On Concrete Mathematics section 5.5, which is teaching the hypergeometric functions, generalized factorials is defined as:
\[
\frac 1 {z!} = \lim_{n \to \infty} \binom{n+z}{n}n^{-z}
\]
where
\[
...
12
votes
6answers
564 views
A question on the Stirling approximation, and $\log(n!)$
In the analysis of an algorithm this statement has come up:$$\sum_{k = 1}^n\log(k) \in \Theta(n\log(n))$$ and I am having trouble justifying it. I wrote $$\sum_{k = 1}^n\log(k) = \log(n!), \ \ ...
9
votes
5answers
645 views
How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$
I'd like a hint to show that:
$$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$
Thanks.
13
votes
6answers
2k views
Stirling's formula: proof?
Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$
Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = ...
11
votes
8answers
992 views
How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$
It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc...
I thought about writing:
$$
a(n) = ...
2
votes
1answer
341 views
What's the limit of the sequence $\lim_{n \rightarrow \infty} \frac{n!}{n^n}$?
$\displaystyle \lim_{n \rightarrow \infty} \frac{n!}{n^n}$
I have a question: Is it valid to use Stirling's Formula to prove convergence of the sequence?
21
votes
1answer
598 views
Repeated Factorials and Repeated Square Rooting
I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square ...
