# Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
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### Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
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### Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
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### Finding the limit of $\frac {n}{\sqrt[n]{n!}}$

I'm trying to find $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$ I tried couple of methods: Stolz, Squeeze, D'Alambert Thanks! Edit: I can't use Stirling.
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### Do factorials really grow faster than exponential functions? [closed]

Having trouble understanding this. Is there anyway to prove it?
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### Why is $i! = 0.498015668 - 0.154949828i$?

While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!. Curiously, Google's ...
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### Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
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### Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is being ...
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### Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?

I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it? Are there any ...
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### Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
10answers
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### What is the purpose of Stirling's approximation to a factorial?

Stirling approximation to a factorial is $$n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.$$ I wonder what benefit can be got from it? From computational perspective (I admit I don't know ...
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### A closed form of $\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right)$

I am looking for a closed form of the following series \begin{equation} \mathcal{I}=\sum_{k=1}^\infty\frac{(-1)^{k+1}}{k!}\Gamma^2\left(\frac{k}{2}\right) \end{equation} I have no idea how to ...
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### Without using a calculator, is $\sqrt{8!}$ or $\sqrt{9!}$ greater?

Which is greater between $$\sqrt{8!}$$ and $$\sqrt{9!}$$? I want to know if my proof is correct... \begin{align} \sqrt{8!} &< \sqrt{9!} \\ (8!)^{(1/8)} &< (9!)^{(1/9)} \\...
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### The product of $n$ consecutive integers is divisible by $n$ factorial

How can we prove that the product of $n$ consecutive integers is divisible by $n$ factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that "...
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Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example: $... 6answers 3k views ###$m!n! < (m+n)!$Proof? Prove that if$m$and$n$are positive integers then$m!n! < (m+n)!$Given hint:$m!= 1\times 2\times 3\times\cdots\times m$and$1<m+1, 2<m+2, \ldots , n<m+n$It looks simple but I'm ... 2answers 12k views ### Is there a way to solve for an unknown in a factorial? I don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1.$n! = \text {a really big number}$Ex.$n! = 9999999$Is there a way to ... 2answers 1k views ### Did I derive a new form of the gamma function? I wish to extend the factorial to non-integer arguments in a unique way, given the following conditions:$n!=n(n-1)!1!=1$To anyone interested in viewing the final form before reading the whole ... 2answers 4k views ### Is$\sqrt{n!}$a natural number? I'm new here (on Mathematics Stack Exchange). Also, I'm a 10th grade student not a math expert. So, my question is whether, $$\sqrt {n!}$$ comes in the set of the Natural Numbers. There were ... 1answer 1k 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 ... 7answers 4k views ### Find$n$, where its factorial is a product of factorials I need to solve$3! \cdot 5! \cdot 7! = n!$for$n$. I have tried simplifying as follows: $$\begin{array}{} 3! \cdot 5 \cdot 4 \cdot 3! \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3! &= n! \\ (3!)^3 \... 3answers 6k views ### Limit of nth root of n! [duplicate] I am asked to determine if a series converges or not: \displaystyle\sum\limits_{n=1}^{\infty} (2^n)n!/(n^n) So I'm using the nth root test and came up with \lim_{n \to {\infty}}(2/n)*(\sqrt[n]{n!})... 1answer 371 views ### Numbers that are clearly NOT a Square Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$The only solutions are ... 2answers 2k views ### Repeatedly taking differences on a polynomial yields the factorial of its degree? Consider a function such that it takes in polynomial function and creates an array of its outputs and then using that array creates another new array by calculating the absolute difference between the ... 4answers 3k views ### Solutions for x!/y!=(y+1)! I was watching a video recently, and I saw how 10*9*8*7 was equal to 7*6*5*4*3*2*1, or to make it clearer, 10!/6!=7!. I was wondering if there were any other solutions, so I checked the web, to find ... 6answers 14k views ### If$n\ne 4$is composite, then$n$divides$(n-1)!\$.

I have a proof and need some feedback. It seems really obvious that the statement is true but it is always the obvious ones that are a little trickier to prove. So I would appreciate any feedback. ...
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### Can n! be a perfect square when n is an integer greater than 1?

Can n! be a perfect square when n is an integer greater than 1? (But is it possible, to prove without Bertrand's postulate. Because bertrands postulate is quite a strong result.)