Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

learn more… | top users | synonyms

51
votes
3answers
2k views

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 ...
48
votes
1answer
2k views

$n!$ 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 ...
44
votes
5answers
4k views

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.
41
votes
11answers
23k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
41
votes
3answers
2k views

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]

Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
40
votes
3answers
3k views

Is $n! + 1$ often a prime?

Related to this question, I wonder how often $n!+1$ is a prime? There is a related OEIS sequence A002981, however, nothing is said if the sequence is finite or not. Does anybody know more about it?
39
votes
14answers
4k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
30
votes
3answers
509 views

Constructing $\mathbb N$ from the set of factorials

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: ...
30
votes
1answer
739 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...
29
votes
1answer
5k views

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 ...
28
votes
3answers
1k views

Which is bigger: $9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$ or $9!!!!!!!!!$?

In my classes I sometimes have a contest concerning who can write the largest number in ten symbols. It almost never comes up, but I'm torn between two "best" answers: a stack of ten 9's (exponents) ...
27
votes
12answers
4k views

$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
27
votes
4answers
3k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
27
votes
2answers
7k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,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 ...
26
votes
12answers
9k views

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 ...
26
votes
4answers
22k views

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 ...
25
votes
6answers
1k 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 ...
23
votes
4answers
1k 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 ...
23
votes
5answers
478 views

How to find $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$?

How to evaluate the following limit? $$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$ For this problem I have two methods. But I'd like to know if there are better methods. My solution 1: Using ...
22
votes
1answer
822 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 ...
21
votes
6answers
7k views

prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer

Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared. My thought process: The numerator is the product of the first n even ...
21
votes
9answers
5k views

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 ...
21
votes
1answer
746 views

are all $n!$ ($n>3$) the difference of two squares?

For the small values of n I have been able to check, it seems that for $n>3$, there exist whole numbers $x,y$ s.t. $n! = x^2 - y^2$. For example .. $4! = 5^2 - 1^2$ $5! = 11^2 - 1^2$ $6! = 27^2 ...
20
votes
1answer
514 views

Integer solutions of $x! = y! + z!$

There was an interesting problem asked about triples $(x,y,z)$ which are solutions of $$x! = y! + z!.$$ Here $(2,1,1)$ is a solution because $2! = 1! + 1!$, as are $(2,1,0)$ and $(2,0,1)$. Now I ...
20
votes
5answers
5k views

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 ...
19
votes
15answers
4k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
19
votes
4answers
667 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
18
votes
5answers
1k views

How best to explain the $\sqrt{2\pi n}$ term in Stirling's?

I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation $$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
18
votes
1answer
901 views

Can a double-factorial be a perfect square?

The title says it, basically. My question is $-$ for $ n \ge 2 $, can $n!!$ be a perfect square, where $!!$ represents the double-factorial? My conjecture is no, but I can't seem to be able to find a ...
17
votes
6answers
3k views

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 = ...
17
votes
3answers
2k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
17
votes
4answers
469 views

Interpreting $n!$ as the volume of a $1 \times 2 \cdots \times n$ box

Q. Are there relationships or proofs that are illuminated by viewing $n!$ as the volume of a $1 \times 2 \cdots \times n$ box in $n$-dimensions? I cannot think of any, but perhaps they ...
17
votes
3answers
450 views

Making $121$ with five $0$s

So I say this puzzle online a few days ago and found it quite interesting. The original question was Make $120$ using only five $0$s. Well, I said to myself, this is utterly trivial. Note that ...
16
votes
3answers
391 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...
15
votes
2answers
5k views

Approximating the logarithm of the binomial coefficient

We know that by using Stirling approximation: $\log n! \approx n \log n$ So how to approximate $\log {m \choose n}$?
15
votes
3answers
214 views

Proving that $\frac{(k!)!}{k!^{(k-1)!}}$ is an integer

I have to prove that: $$\frac{(k!)!}{k!^{(k-1)!}} \in \Bbb Z$$ for any $k \geq 1, k \in \Bbb N$ Tried doing $t = k!$ which would give $$\frac{t!}{t^{t/k}}$$ But I think I just made it harder, and ...
14
votes
13answers
3k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
14
votes
4answers
3k views

Factorial of infinity

So, I've read in this article that: $$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$ And that: $$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$ I found this result very ...
14
votes
2answers
367 views

On the Limit of Stirling's Approximation

I have recently proven the following curious identity: For real $x \geqslant 1$, \begin{align} \lfloor x \rfloor! = x^{\lfloor x \rfloor} e^{1-x} e^{\int_{1}^{x} \text{frac}(t)/t \ dt} \end{align} ...
13
votes
6answers
1k views

Why $0!$ is equal to $1$? [duplicate]

Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my ...
13
votes
6answers
875 views

A question on the Stirling approximation, and $\log(n!)$

In the analysis of an algorithm this statement has come up:$$\sum_{k = 1}^n\log(k) \in \Theta(n\log(n))$$ and I am having trouble justifying it. I wrote $$\sum_{k = 1}^n\log(k) = \log(n!), \ \ ...
13
votes
8answers
5k views

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

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) = ...
13
votes
1answer
368 views

Yet another $\sum = \pi$. Need to prove.

How could one prove that $$\sum_{k=0}^\infty \frac{2^{1-k} (3-25 k)(2 k)!\,k!}{(3 k)!} = -\pi$$ I've seen similar series, but none like this one... It seems irreducible in current form, and I have ...
12
votes
7answers
7k views

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 ...
12
votes
4answers
3k views

Solve by induction: $n!>(n/e)^n$

To Prove : $n! > (n/e)^n$ The question seems easy but it ain't; anyone up for it ?
12
votes
1answer
146 views

Solving $n!+m!+k^2=n!m!$ for positive integers $n,m,k$

I have been running in circles with this for a while now. It seems that the only solution is $(n,m,k)=(2,3,2)$ but I don't know how to prove it. Things I have noticed: WLOG $n\geq m$ we see that ...
11
votes
2answers
801 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
11
votes
4answers
16k views

Factorial, but with addition [duplicate]

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 ...
11
votes
3answers
797 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...
11
votes
1answer
73 views

Does it follow that $(n!)^n$ divide $(n^2)!$

It is well known that $(n!)^2$ divides $(2n)!$. Does it follow that $(n!)^3$ divides $(3n)!$ and so on up to $(n!)^n$ dividing $(n^2)!$? If yes or no, could you provide the details behind the ...