# 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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### Closed form of basic hypergeometric series: Adapt the answer given in linked question to solve my own variation?

I'm interested in two things: A computationally efficient (used here to mean the number of terms is bounded and not dependant on the size of the input) partial sum formulae of the expression below, ...
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### How to prove $\frac{m!}{n!} \geq n^{m-n}$

How to prove the following: $$\frac{m!}{n!} \geq n^{m-n}$$ In my book it's written: "easy to prove by separately considering the cases $m \geq n$ and $m<n$). I tried using the bounds of ...
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### Need help showing Riemann's Functional equation for negative numbers and complex numbers

Riemann's Functional equation: $\zeta(-z)$=${-2*z!\over(2\pi)^{z+1}}$$sin({\pi z\over2})$$\zeta(z+1)$This formulas expresses $\zeta(-z)$ in terms of $\zeta(z+1)$ Note: I read that the author said, ...
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### Can the sum of the first $p$ factorials ever be a perfect power for $\ p>3\$?

Has $$\sum_{j=1}^p j!=q^r$$ , where q,p,r are positive integers, and r > 1 , a solution ? I solved partially, if r is even, then RHS is a perfect square, and there is no doubt in that. Therefore, the ...
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### Finding sum to infinity: $\sum\limits_{n = 1}^{ \infty}\frac{n^2}{n!}$ [duplicate]

I am trying to find what this value will converge to $$\sum_{n = 1}^{ \infty}\frac{n^2}{n!}$$ I tried using the Taylor series for $e^x$ but couldn’t figure out how to manipulate it to get the above ...
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### $(\frac{n}{e})^{n} < n! < (\frac{n}{e} + n\varepsilon)^{n}$ doesn't comply with the limit definition?

I try to understand what I've overlooked, when I came up with this inequality: First, we have this limit: $$\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{n^n}} = \frac{1}{e}$$ Which gives, by the ...
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### Longest sequence of consecutive integers which are not coprime with $n!$

For any integer $n$, the factorial $n!$ is the product of all positive integers up to and including $n$. Then in the sequence $$n!+2,n!+3,... ,n!+n$$ the first term is divisible by $2$, the second ...
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### Check if $k$ is divisible by $2^9$ or $2^{10}$

$k=\frac{512!}{256!*128!*...*2!*1! }$ I need to check if the expression k is divisible by $2^9$ or $2^{10}$. This is a multiple choice question and the options and the question goes like this: ...
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### Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N??

Can x mod (N - a) or x mod (N + a) be calculated just by knowing x mod N? where a is an arbitrary integer, and N is a prime. e.g. i know 22! mod 23 ≡ 22 using Wilson's theorem. lets say i want to know ...
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### Can't find a seemingly simple limit $\lim_{n\to\infty}\frac{(n+k)!}{n^n}$

Evaluate the limit: $$\lim_{n\to\infty}\frac{(n+k)!}{n^n}, \ n,k\in\Bbb N$$ I would like to avoid Stirling's approximation, derivatives and Cesaro-Stolz, since none of them has been yet ...
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### A question on the approximation of $\ln(x!)$ for small x

In a question of Arfken and Weber's Mathematical Methods for Physicists, For small values of $x$, $\ln(x!)=-\gamma x+\sum\limits_{n=2}^{\infty}(-1)^n\frac{\zeta(n)}{n}x^n$, where the symbols used ...
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### Show that $\frac{(2n)!!}{(2n+1)!!}$ converges.

Given a sequence $\{x_n\}, \ n\in\Bbb N$: $$x_n = \frac{(2n)!!}{(2n+1)!!}$$ Show that $x_n$ converges. I'm wondering why I'm getting a seemingly wrong result (assuming the problem statement ...
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### Determining the missing digits of $15! \equiv 1\square0767436\square000$ without actually calculating the factorial

$$15! \equiv 1\cdot 2\cdot 3\cdot\,\cdots\,\cdot 15 \equiv 1\square0767436\square000$$ Using a calculator, I know that the answer is $3$ and $8$, but I know that the answer can be calculated by ...
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### Methods for proving a function outputs an infinite number of integers

I have a function involving polynomials and the centre of the Binomial Triangle and I'd like to prove that the function produces a positive integer infinitely many times. I don't have any interest in ...
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### Probability an ace lies behind first ace

Consider a deck of 52 cards. I keep drawing until the first ace appears. I wish to find the probability that the card after is an ace. Now, the method I know leads to the correct answer is that given ...
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### Number of divisors of $10!$
Determine the amount of divisors of $10!$ This is a question in my combinatorics textbook, so I need to somehow reduce this to an elementary counting problem like combinations, permutations with or ...