# 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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### 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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### Highest power of a prime $p$ dividing $N!$

How does one find the highest power of a prime $p$ that divides $N!$ and other related products? Related question: How many zeros are there at the end of $N!$? This is being done to reduce abstract ...
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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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### 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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### How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$

I'd like a hint to show that: $$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$ Thanks.
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### What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?

$$\lim_{n \to\infty} \frac{n!}{n^n}$$ I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence?
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### Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. Here's the formula: $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$ Can anyone give a proof of this result? Note: ...
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### To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k , k\in \mathbb Z^+$ ?
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### How to find the factorial of a fraction?

From what I know, the factorial function is defined as follows: $$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$ And $0! = 1$. However, this page seems to be saying that you can take the factorial of a fraction,...
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### Why does 0! = 1? [duplicate]

Possible Duplicate: Prove $0! = 1$ from first principles Why does $0! = 1$? All I know of factorial is that $x!$ is equal to the product of all the numbers that come before it. The product of 0 ...
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### How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
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### Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 ...
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### Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in terms ...
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### 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 ...
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### How many zeroes are in 100!

One common math puzzle I've seen around asks for how many zeros are in the product of "100!" Usually, the solution everyone gives goes something like try to match pairs of 5s and 2s that factor out ...
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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.)
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### Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$ [duplicate]

I am trying to calculate the following limit without Stirling's relation. $$\lim_{n \to \infty} \dfrac{n!}{n^n}$$ I tried every trick I know but nothing works. Thank you ...
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### Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the sum ...
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### Proving $\sum_{k=1}^n k k!=(n+1)!-1$

Prove: $\displaystyle\sum_{k=1}^n k k!=(n+1)!-1$ (preferably combinatorially) It's pretty easy to think of a story for the RHS: arrange $n+1$ people in a row and remove the the option of everyone ...
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### Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$\sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
### To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 … (2n-1)\over 2.4.6…(2n)$
How can I show $\lim_{n \to \infty} a_n = 0$ $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$ I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below ...