Tagged Questions

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 ...
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. ...
70 views

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? $$\lim_{n \to +\infty} \frac{2^{n}}{n!}$$
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... ...
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 ...
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 ...
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: ...
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 ...
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 ...
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 \[ ...
564 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?