# Tagged Questions

Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
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### What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
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### How can I use Stirlings inequality to prove this inequality?

Let $p,k$ be natural numbers with $p\ge k$, show that $$\frac{(p-k)!}{(p+k)!} \le \frac{1}{p^{2k}} \left(\frac{e}{2} \right)^{2k^2/p}.$$ The text where I come across this says to use Stirling's ...
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### If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's formula.....
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### Solving equation involving factorials

I have this particular equation $\frac{(\alpha-1)!(\beta-1)!}{(\alpha+\beta-1)!} = \frac{\Gamma(p)(1+q)^{n+2p} 2^n}{q^{p}(2+q)^{n+p}}$. Now, given the values of $\alpha$ and $\beta$, I need to find ...
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### The inverse of x!

what is the inverse of a factorial function? Its is not continuous but is modeled by the gamma function which is continuous so must have a inverse any research leads to the inverse gamma function that ...
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### Sum problem involving Factorials

got the following problem to prove for $n \in \mathbb{N}$ and $1 \leq i \leq n$: \sum_{k=1}^{n-i} \frac{(2n-1-i-2k)! 2^{2k}}{(n-i-k)! (n-k)! 2}=\sum_{k=1}^{n-i} \frac{(2n-1-i-k)! 2^{...
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### Find k such that, $\displaystyle\frac{n^{\underline k}}{n^k}<a$

How to find k here $\displaystyle\frac{n^{\underline k}}{n^k}<a$ ? With, $n^{\underline k}=n\cdot(n-1)\cdot...(n-k+1)$ and $(n>k,\ a>0)$ Of course if $k\approx n/2$ the inequality ...
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### Transforming a Riemann-Stieltjes integral related to the factorial

I have been able to show that $$\log n! = \int_{1 + \epsilon}^n \log x \, d\lfloor x \rfloor$$ but I have not been successful trying to transform this Riemann-Stieltjes integral to an ordinary ...
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### Find all possible solutions!

Find solutions for $$^nP_r=s!$$ For $(n,r,s)\in \mathbb{N}$ I could find some trivial solutions $(6,3,5)~,~(1,1,1)$ etc.
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### Maxima and minima of $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}$.

Find the maximum and minimum value of the function $f(x)=\binom{16-x}{2x-1}+\binom{20-3x}{4x-5}.$ ATTEMPT:- By A.M.-G.M. inequality, $\frac{a+b}{2}\ge\sqrt{ab}$, $\quad$ for $a,b\gt 0$ with equality ...
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### Factorial ratio sum of finite series

Given: $S = \sum_{i=1}^{n-1}{i! \over n!}$ How would I find the sum for an arbitrarily large $n$ ? Example: $n=5$ $S = \frac{1!}{5!} + \frac{2!}{5!} + \frac{3!}{5!} + \frac{4!}{5!} = 0.275$
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### Limit of the spherical Bessel function of the second kind

I know that the limit for the spherical bessel function of the first kind when $x<<1$ is: $j_{n}(x<<1)=\frac{x^n}{(2n+1)!!}$ I can see this from the formula for $j_{n}(x)$ (taken from ...
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### Permutations to divide a solid

I have a 3-dimensional cuboid, with dimensions 2x2x1. I wish to divide this into smaller EQUAL sized cuboids of size 2x1x1. Then I want to extend this case for larger cuboids, assuming that equal ...
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### Explaination of how Wallis's formula was used to arrive at the end result?

While looking for a proof on Stirling's formula I am having a little trouble connecting the dots between these last two steps here: [1]: ! The first formula is Wallis's formula.
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### Factor out factorial from expression

I have the expression $(k+1)! - 1 + (k+1)(k+1)!$ How do I factor out $(k+1)!$ to achieve the result: $[(k+1)!(k+2)] - 1$? I for the life of me cannot figure this out. Thanks!
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### Product of complex numbers $m+in$ with $0 < m,n \leq N$

I am trying to look for a generalization of Stirling's formula to complex numbers. In the integer case: $$\log n! = \sum_{k = 1}^n \log k \approx \int_1^n \log x \, dx = n \log n - n$$ For the ...
In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...