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

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7
votes
6answers
155 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
4
votes
0answers
42 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
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 ...
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 ...
1
vote
0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
3
votes
1answer
34 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
2
votes
2answers
44 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
4
votes
2answers
295 views

Exponent of Prime in a Factorial [duplicate]

I was just trying to work out the exponent for $7$ in the number $343!$. I think the right technique is $$\frac{343}{7}+\frac{343}{7^2}+\frac{343}{7^3}=57.$$ If this is right, can the technique be ...
2
votes
3answers
168 views

Exponent of $p$ in the prime factorization of $n!$

Exponent of $p$ in the prime factorization of $n!$ is given by $\large \sum \limits_{i=1}^{\lfloor\log_p n \rfloor } \left\lfloor \dfrac{n}{p^i}\right\rfloor $. Can this sum be simplified further to ...
6
votes
3answers
423 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
1
vote
2answers
61 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 ...
-1
votes
1answer
10 views

Common factors /greatest common factors /least common factor

See I am baffling with the concept of above mentioned title I saw one question they wrote find common factors /gcf/lcm of 42,294,882 Please help me out in getting the concept Thanks
3
votes
1answer
104 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
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 ...
0
votes
1answer
37 views

How do I prove the formula for multichoose?

In combinatorics, there is a formula "$n$ multichoose $k$", which is the way of making a multiset having $k$ elements choosing out of $n$ options. "$n$ multichoose $k$" is the same as "$(n+k-1)$ ...
1
vote
1answer
51 views

Approximating $\frac{(kn)!}{(n!)^k}$

Is there any approximations for the form $$\frac{(kn)!}{(n!)^k},$$ where $n$ and $k$ are positive integers? $n$ is not necessary much larger than $k$?
0
votes
0answers
36 views

How to write $n!=a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$?

We know that $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1, n\in \mathbb N$. Now I am willing to write $n!$ as $a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$ where $a, r, \alpha\in \mathbb ...
2
votes
2answers
357 views

How to prove this inequality with factorials: $n!>n^{\frac {n}{2}}$ [duplicate]

This is what I am trying to solve: $n!>n^{\frac {n}{2}}$ I tried with induction and somehow it doesn`t work this way, I tried with logarithms and also didn´t find a way. Is there an elementary(if ...
0
votes
3answers
56 views

How do I derive $n!$ from this series?

I am reading a book where the following reduction is performed, but it's not explained exactly what is going on. I'm sorry if this is a dumb question, but I simply don't get how we are deriving the ...
8
votes
4answers
141 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
6
votes
0answers
74 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
1
vote
2answers
44 views

How do I evaluate this $ \lim \frac{\zeta(n)} {({n)!}} , n\to\infty $ if it was existed?

Is there someone who can show me how do I evaluate this limit $$ \lim_{n\to +\infty} \frac{\zeta(n)} {n!} $$ if it exists ? Thank you for any help.
0
votes
1answer
59 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 $$\sum _{ n=1 }^{ \infty }{ \frac { 1 }{ \left( n+2 \right) n! } } $$
3
votes
3answers
170 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
1
vote
2answers
67 views

How can I show that $n^{n+2}<(2n)!$ for any integer $n$.

When I was try to show that the series $$\sum_{n\in\mathbb{N}} \frac{n^n}{(2n)!}$$ is convergent using comparison test, I stuck at the point $$n^{n+2}<(2n)!$$ I think it can be show using ...
6
votes
1answer
88 views

Why is $!0 = 1$?

The subfactorial function is defined as: $$!n = n!\sum_{i=0}^n\dfrac {(-1)^i} {i!}$$ I was curious and wanted to find out what $!0$ came out to be. Since I couldn't use it in the sum above, I used a ...
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 ...
0
votes
2answers
47 views

Simplifying expressions with factorials

If $$\large{a_n = \frac{x^n}{2^n n!}}$$ , Then find $$\large{ \frac{a_{n +1}}{a_n}}$$ .. I tried the following: $$\large{a_{n + 1} = \frac{x^n}{2^n n!} + \frac{2^n n!}{ 2^n n!} = \frac{x^n + 2^n ...
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?
7
votes
4answers
158 views

What is the limit as k approaches infinity of $(k!)^{\frac{1}{k}}$ [duplicate]

What is the value of $$\lim_{k\to\infty}(k!)^{\frac{1}{k}}?$$ One of my students concluded the limit was infinity – which I tend to agree with, but was unable to show that was the limit. We ...
10
votes
6answers
251 views

Determine whether $\frac{1000!}{100!^{10}}$ is an integer

Can you give an idea, how to find out whether the result of ${1000!}/{100!^{10}}$ an integer. Modulo division? But what I met was about powers like $2^{100}/125$...
4
votes
3answers
103 views

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $? [duplicate]

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $ ? I tried taking using logarithm to bring the expression to sum form and then tried L Hospital's Rule.But its not working.Please ...
1
vote
2answers
61 views

How do I find the probability of some elements being together inside a randomly arranged set?

If I have a total of $n$ balls made of $k$ red balls and $(n-k)$ green balls and I arrange them all randomly in a line, how can I calculate the probability $x$ of a group of $y$ red balls being ...
2
votes
2answers
61 views

Equal products of consecutive integers

We have that $1\cdot2\cdot3\cdot4\cdot5\cdot6=8\cdot 9\cdot10$. An easy consequence is that $7!=7\cdot8\cdot9\cdot10$. I have been looking for more non trivial examples like these, but I have found ...
2
votes
1answer
66 views

Why do the closest primes whose distance $d \gt 1$ to $c(n)=\frac{(n+1)!+n!}{2}$ have always $d \in \Bbb P$?

I have made the following observation: define the center of $n!$ and $(n+1)!$, $c(n)$, as the number located exactly in the middle of $(n+1)!$ and $n!$. Def: $\forall n \gt 2\ , \ ...
1
vote
1answer
34 views

Proving that $\Delta x^{(n)} = n x^{(n-1)}$

Define $\Delta f(x) = f(x+1) - f(x)$ (the difference operator). Define $x^{(n)} = x(x-1) \dots (x-n+1)$ (the falling factorial function). There's a rather simple theorem which shows that $\Delta ...
0
votes
1answer
39 views

What function does Wolfram Alpha plot instead of the factorial?

Look to the second graph where Wolfram Alpha gives a continuous factorial function: What is the second graph? It is not the gamma function, since that has $\Gamma(-1)=0!=1$.
-3
votes
1answer
87 views
2
votes
5answers
6k views

Calculating the limit $\lim((n!)^{1/n})$

Find $\lim((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm, $$\lim \ln((n!)^{1/n}) = ...
9
votes
2answers
72 views

When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

As stated in the title: when is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $m,j,n$? I was thinking about this problem a couple of days ago because in all my years of ...
1
vote
1answer
55 views

Number of Divisors of N factorial

Say d(N) = Number of factors of N! Briefly: I wish to know if there is a Recurrence relation for this problem Now I wish to Know if there is a way to calculate d(N) in terms of previously calculated ...
1
vote
1answer
80 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation ...
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.
7
votes
2answers
144 views

Show that $\lim\limits_{n \to \infty} \frac{(n!)^{1/n}}{n}= \frac{1}{e}$ [duplicate]

Show that $$\lim_{n \to \infty} \left\{\frac{(n!)^{1/n}}{n}\right\} = \frac{1}{e}$$ What I did is to let $U_n = \dfrac{(n!)^{\frac{1}{n}}}{n}$ and $U_{n+1} = ...
2
votes
2answers
573 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
0
votes
1answer
24 views

Find number of possible combinations from list of items [closed]

I have the following list: Product Icecream Banana Strawberry Vanilla The possible combinations (based on the spec) are 5: ...
2
votes
1answer
48 views

Total possible ways of representing n! as a sum of two or more consecutive positive integers.

I need to calculate total possible ways of representing $n!$ as a sum of two or more consecutive positive integers. Example : $3!=1*2*3=6$ and $6=1+2+3$ the only one possible way. Answer : $1$ The ...
2
votes
2answers
87 views

Total number of divisors of factorial of a number

I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where ...
0
votes
3answers
31 views

Permutations and Combinations equation [closed]

I cannot figure out how I am supposed to start in solving for $n$ in this equation. $$_nP_4=84(_nC_2)$$ Thank you!
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 ...