Mathematics Stack Exchange
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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.

The factorial is defined as the product of all positive integers less than or equal to some integer $n$, written $n!$. Multiple $!$'s skip integers, so for example $10!!!=10\cdot7\cdot4\cdot1$.

This function is only defined over integers, but the extends it to all complex numbers that are not non-positive integers.

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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 sense, with $$\mathsf{A} = \left(\begin{matrix} 1…
Laplacian
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Is there a function that grows faster than exponentially but slower than a factorial?

In big-O notation the complexity class $O(2^n)$ is named "exponential". The complexity class $O(n!)$ is named "factorial". I believe that $f(n) = O(2^n)$ and $g(n) = O(n!)$ means that $\dfrac{f(n)}{g(n)}$ goes to zero in the limit as $n$ goes to…
Robert L. Read
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$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and $2n-2$. A proof can be found here. Two weeks and…
Lincoln Blackham
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Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
Billy Thompson
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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.
user6163
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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.
user 1591719
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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 just because of the way it's defined.
Ali Abbasinasab
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Why does "Turn! Turn! Turn!" equal 241217.524881?

If you search for "Turn! Turn! Turn!" on Google, then the second result is this YouTube video of The Byrds performing the Pete Seeger song of that name. But the first result is Google's internal calculator displaying "241217.524881". With a bit of…
tparker
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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 repurposed in an effort to cut down on duplicates,…
Matt Nashra
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Factorial, but with addition

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the factorial $5!$ way. EDIT: I know about the formula. I want to…
akinuri
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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 function. What use might it be to take the factorial of…
Oliphaunt
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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 $C = \sqrt{2 \pi}$. What is a good way of doing this? Could…
James
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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 calculator dutifully informed me that $i!$ was, in fact, $0.498015668 - 0.154949828i$. Why…
growse
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Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
Ssegawa Victor
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$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for the series: $$ 1 + 2 + 3 + 4 + \cdots + n =…
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