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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9
votes
2answers
25k views

Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq N$...
28
votes
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 ...
31
votes
8answers
3k views

Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$

Let $m$ be a positive integer and $n$ a nonnegative integer. Prove that $$(n!)\cdot(m!)^n|(mn)!$$ I can prove it using Legendre's Formula, but I have to use the lemma that $$ \dfrac{\...
25
votes
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 ...
13
votes
3answers
7k views

For all $n>2$ there exists a prime number between $n$ and $ n!$

How to prove that there exists a prime number between $n$ and $ n!$, for all $ n> 2$? (Bertrand's postulate gives a much better bound, but this question is about obtaining a self-contained proof.)
4
votes
1answer
252 views

Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$

Find the limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}.$ The limit is supposed to be $e^{-1}$ and there are a couple of posts in MSE proving it. But here is a proof I encountered showing ...
7
votes
6answers
724 views

Evaluate $\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
6
votes
2answers
5k views

Number of zeroes at end of factorial

Question: How many zeroes will there be at the end of $(127)!$ Approach: Considering the fact that when two numbers ending in $x$ and $y$ zeroes are multiplied, the resulting number contains $x+y$ ...
3
votes
2answers
245 views

Prove $\sum_{i=1}^n i! \cdot i = (n+1)! - 1$?

Prove the summation: $$\sum_{i=1}^n i! \cdot i = (n+1)! - 1$$ using induction. base case: $n=1$: \begin{align*} \sum_{i=1}^1 i! \cdot i &= (1+1)! - 1 \\ 1 &= 2 - 1 \\ 1 &= 1 \end{align*...
6
votes
3answers
216 views

Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$ [duplicate]

$$\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$$ Now, $\log(n!) = \Theta (n\log(n))$ so I think we could write, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - \log\left(...
4
votes
2answers
218 views

$\sum_{i=1}^n i\cdot i! = (n+1)!-1$ By Induction

I am trying to prove the following by Mathematical Induction: $$\sum_{i=1}^n i\cdot i! = (n+1)!-1\quad\text{for all integers $n\ge 1$}$$ My proof by Induction follows: First prove $P(1)$ is true, $...
6
votes
3answers
10k views

Find $\lim_{n \to \infty} \sqrt[n]{n!}$.

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
4
votes
3answers
119 views

Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$

I am trying to calculate the following limit $$\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$$ where $k$ can be any number between $0$ and $N$. I thought of the following: If I take the logarithm of the ...
7
votes
7answers
404 views

How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$

So guys, how can I evaluate and prove that $$\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0.$$ Any ideas are welcomed. $n!!$ is the double factorial, as explained in this wolfram post.
15
votes
8answers
19k views

How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$ [duplicate]

It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc... I thought about writing: $$ a(n) = \frac{...
10
votes
1answer
438 views

Solutions of $p!q! = r!$

The title says it all, more or less. Obviously, there are infinitely many "trivial" integral solutions of the form $p=n, q=(n!-1), r= n!$. How many non-trivial solutions are there? I came across this ...
5
votes
2answers
2k views

Finite Summation of Fractional Factorial Series

Is there a closed form solution for the following series? (Without Using Gamma Function): $$ S=\sum _{i=1}^{n-1} \frac{1}{(i+1)!} $$
10
votes
5answers
790 views

Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]

Possible Duplicate: Summation of a factorial This equation is given: $$ 1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1 $$ I've solved it using mathematical induction but I'm ...
9
votes
4answers
185 views

Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
8
votes
3answers
10k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$ [duplicate]

$$1 + \sum_{j=1}^{n} j!j$$ I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
7
votes
6answers
170 views

I need help to advance in the resolution of that limit: $ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $

how I can continue this limit resolution? The limit is: $$ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $$ This is that I have done: I apply this test: $ \lim_{n \to \infty}{\sqrt[n]{a_n}} = \...
7
votes
2answers
473 views

Closed form for $(p-n)!\pmod{p}$ where $p$ is prime

Does $(p-n)!\pmod{p}$ have a closed form for any $n>2$ when $p$ is prime? $(p-0)!=0 \pmod{p}$ $(p-1)!=-1\pmod{p}$ $(p-2)!=1\pmod{p}$
8
votes
3answers
500 views

Is $\sum\limits_{n=1}^{\infty}\frac{n!}{n^n}$ convergent?

I can clearly see that $\dfrac{n!}{n^n}\to 0$ when $n\to\infty$. But how do I know if the sum $$\sum_{n=1}^{\infty}\frac{n!}{n^n}$$ is convergent or not? I know this might be basic, but thank you if ...
6
votes
2answers
4k views

How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?

I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these. First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = 12$ ...
4
votes
4answers
2k views

Mathematical problem induction: $\frac12\cdot \frac34\cdots\frac{2n-1}{2n}<\frac1{\sqrt{2n}}$

This is a problem that I tried to solve and didn't come up with any ideas .?$$\frac{1}{2}\cdot \frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n}}.$$ All I get is $\frac{1}{\sqrt{2n}}\cdot\frac{2n+...
1
vote
3answers
4k views

limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
0
votes
1answer
177 views

Determine the value of the series $\sum _{ n=1 }^\infty \frac1{ ( n+2) n!}$

Find the partial sum $S_n$ of the telescoping series $$\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ \left( n+2 \right) n! } } $$
8
votes
3answers
965 views

Show this equality (The factorial as an alternate sum with binomial coefficients).

Why does the following equality hold? $$n!=\sum_{k=1}^n (-1)^{n-k} \binom{n}{k} k^n$$
13
votes
5answers
1k views

$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction

I have to show $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction. This is where I am stuck: $$\left( \frac{n+2}{2} \right)^{n+1} \geq \dots \geq =2 \left( \frac{n+1}{2} \right)^{n+1} = \...
11
votes
3answers
13k views

How to define fractional factorials, like 3.6!? [duplicate]

I did not know that you could find an answer for that. However, I can only use Excel so far to do it. How to calculate 3.6! by hand?
5
votes
2answers
683 views

Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$ n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this formula ...
4
votes
3answers
5k views

Proving that $n!≤((n+1)/2)^n$ by induction [duplicate]

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): (n+1)!≤((n+2)/2)^{(n+...
6
votes
2answers
286 views

Understanding a very elementary property of factorials

I've seen this stated in a few places. If $$\vartheta(x) = \sum_{p\le{x}} \log (p) \qquad \psi(x) = \sum_{m=1}^{\infty}\vartheta\left(\sqrt[m]{x}\right)$$ Then $$\log(x!) = \sum_{m=1}^{\infty} \...
4
votes
3answers
260 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
3
votes
3answers
269 views

Summation of a series help: $\sum \frac{n-1}{n!}$

Can someone teach me how to solve the following series. I have no idea how to deal with factorial. $$\sum_2^\infty \frac{n-1}{n!}$$ Thanks
2
votes
3answers
2k views

How to show that $\gcd(n! + 1, (n + 1)! + 1) \mid n$?

Let $n$ be a positive integer, $n!$ denotes the factorial of $n$. Let $d = \gcd(n! + 1, (n + 1)! + 1)$. Show that $d$ divides $n$. (Hint: notice that $(n+1)(n!+1) = (n+1)!+n+1$)
1
vote
5answers
299 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
13
votes
4answers
2k views

If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
7
votes
1answer
149 views

Calculating $\sum_{k=1}^nk(k!)$ combinatorially [duplicate]

The sum $\sum_{k=1}^nk(k!)$ can be easily calculated by noting $k(k!)=(k+1)!-k!$. Is there a way to calculate the sum nicely using a combinatorial argument. Is it possible to notice it is $(n+1)!-1$ ...
4
votes
0answers
254 views

Why does $\lim_{n\to\infty} \frac{n!}{(n-k)!n^k}$ equal 1 [duplicate]

Came across this in my maths book. $$\lim_{n\to\infty} \frac{n!}{(n-k)!n^k} = 1$$ Why is it so?
87
votes
13answers
95k views

Do factorials really grow faster than exponential functions? [closed]

Having trouble understanding this. Is there anyway to prove it?
22
votes
4answers
8k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac 1 {2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdots \infty = \infty! = \sqrt{2\pi}$$ I found this result very strange and ...
20
votes
5answers
1k views

Prove $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
7
votes
4answers
313 views

Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

$$n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$$ Could anyone give the proof of the above equation? Thanks in advance!
10
votes
4answers
312 views

Determine $4$ specific digits in $34!$

Find the values of $a,b,c,d\in\mathbb{N}$ such that $$ 34!=295232799cd9604140847618609643ab0000000 $$ My Attempt: The factorial of $34$ contains a $3$, so the RHS must be divisible by $3$. ...
12
votes
2answers
1k views

Ramanujan's approximation to factorial

I saw this approximation for the factorial given by Ramanujan as $$\log(n!) \approx n \log n - n + \frac{\log(n(1+4n(1+2n)))}{6} + \frac{\log(\pi)}{2}$$ in wikipedia, which claims the approximation is ...
8
votes
3answers
46k views

Any shortcut to calculate factorial of a number (Without calculator or n to 1)?

I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...
8
votes
2answers
390 views

Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer

I need help to prove that the quantity of this division : $\dfrac{({10!})!}{{10!}^{9!}}$ is an integer number, using combinatorial proof
16
votes
4answers
4k views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
16
votes
5answers
572 views

Finding all the numbers that fit $x! + y! = z!$

I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it. ...