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

1
vote
2answers
53 views

Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
0
votes
1answer
38 views

Summation with factorial

I want to understand how this step is performed. Can you tell me that how this value of Po is obtained from the first equation.! ...
1
vote
6answers
107 views

Is $\frac{n}{3}! = (\frac{1}{3})^n n!$

Is $$\frac{n}{3}! = (\frac{1}{3})^n n!$$ I thought I could take all the (1/3) out of the factorial, but wolfram alpha says this is false.
1
vote
4answers
85 views

$\lim_{ n\to \infty} \frac{n!}{n^n}$ via L'Hospital's rule

I just need to find this limit and I don't know how to use L'Hopital's rule in this case: $$\lim_{ n\to \infty} \frac{n!}{n^n}.$$ I apologize for the lack of formatting, I've never used the site ...
0
votes
2answers
37 views

Prove $n!\mid\prod_{k=i}^{i+n-1}k$

I have no idea how to prove this, I haven't yet learned much about this kind of product. $$ n!\mid\prod_{k=i}^{i+n-1}k $$
0
votes
1answer
33 views

Proving two equations containing ceiling and floor function to be equal

The problem is to show that if $n$ is a positive integer, then $$1 = C(n, 0) < C(n, 1) < ... < C(n, \left \lfloor{ n/2} \right \rfloor) = C(n, \left \lceil {n/2} \right \rceil) > ... > ...
2
votes
1answer
54 views

Wonder how to evaluate this factorial $\left(-\frac{1}{2}\right)!$

I've learned factorial. But today I saw a question which I don't know how to start with: $$\left(-\frac{1}{2}\right)!$$ Can anyone explain how to solve it? Thanks
2
votes
8answers
133 views

How can $0!=1$ if the definition of factorial is $n!=n\times (n-1)!$ [duplicate]

Its a pretty basic question. If the definition of factorial is $n!= n\times(n-1)!$, then how can $0!=1$ since if we feed $0$ into the equation we get $0!=0\times (-1)!$? This comes after a ...
2
votes
2answers
79 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 ...
1
vote
3answers
41 views

How to calculate $\lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$

I need to calculate limit number 1, and I don't understand how to get out the factors. $$ (1) \lim_{k \to \infty} \frac{(2k)!}{(2k+2)!}$$ $$ (2) \lim_{k \to \infty} \frac{(k)!}{(k+1)!}$$ When I ...
0
votes
1answer
34 views

function to approximate $x!$ without factorial

I am looking for a function $f(x)$ such that $f(x)\approx x!$, but (obviously) the function of x does not use factorial, eg a polynomial or exponential function. it does not have to be precise, just ...
0
votes
0answers
28 views

Coefficients for the falling factorial

Hello fellow mathematicians, I am trying to find a generating function, or at least find some useful property from the coefficients of the falling factorial. Let $(x)_n$ denote a falling factorial, ...
2
votes
1answer
41 views

integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$

Total no. of integer ordered pair of $(x,y,z)$ in $x!+y! = z!.\;,$ Where $x,y,z\in \mathbb{W}$ $\bf{My\; Try::}$ Let $w=\max\left\{x,y\right\}$. Then $w<z$. So we can write $w\leq (z-1)$ So ...
2
votes
1answer
38 views

Proof for 1/k! using n choose k as n approaches infinity and its relation to the gamma function

Prove that $\lim_{ n \to \infty }\binom{n}{k}(1/n)^k =\frac{1}{k!}$ How is this related to the gamma function?
4
votes
1answer
74 views

Combinatorial interpretation of double factorial.

Using some basic algebra (and proved afterwards using induction), I found that: $$ 1 \cdot 3 \cdot ... (2n-1) = \frac{(2n)!}{2^n \cdot n!}$$ After a bit of research, I found out that this is known ...
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 ...
2
votes
2answers
50 views

How can I express the ration of double factorials $\frac{(2n+1)!!}{(2n)!!}$ as a single factorial?

How can I change the double factorial of $$\frac{(2n+1)!!}{(2n)!!}$$ to single factorial?
0
votes
2answers
43 views

Show that $n! < (n/2)^n$ for all large enough $n$ in as elementary a way as possible

Show that $n! < (n/2)^n$ for large enough $n$ in as elementary a way as possible. Using Stirling's formula is not allowed. Of, course, what is true, is that $n! < (n/c)^n$ for any $c < e$ ...
3
votes
1answer
38 views

Expressing $\sum_{k=1}^{n}\frac{1}{(k+2)k!}$ in terms of $n$.

How would I express $$\sum_{k=1}^{n}\frac{1}{(k+2)k!}$$ in terms of $n$? An attempt of mine is $$\sum_{k=1}^{n}\frac{1}{(k+2)k!} = \sum_{k=1}^{n}\frac{1}{(k+1)! + k!},$$ which is not useful for ...
2
votes
1answer
95 views

Factorials…How do they do it?

So I've been recently arguing with my teacher about factorials. My teacher says that factorials can only be calculated for integers, because the definition of factorials is as follows: the product ...
2
votes
2answers
66 views

Summation over a product of binomial coefficients

Question: I can't figure out why the following equality is true $\sum_\limits{k=a-b-c}^{d} (-1)^k \binom{d}{k}\binom{k+b+c}{a} = (-1)^d \binom{b+c}{a-d} $ How can this be shown? (In the book it just ...
6
votes
2answers
106 views

Sum of factorial fractions

Find the sum $$\sum\limits_{a=0}^{\infty}\sum\limits_{b=0}^{\infty}\sum\limits_{c=0}^{\infty}\frac{1}{(a+b+c)!}$$ I tried making something like a geometric series but couldn't. Then I couldn't think ...
2
votes
1answer
70 views

Floor function of a factorial

Compute $$\left\lfloor \frac{1000!}{1!+2!+\cdots+999!} \right\rfloor.$$ How can I start with the problem? I thought of dividing by some number, but then I thought that some small numbers when added ...
-2
votes
1answer
49 views

Inverse question of trailing zeros [duplicate]

$(5n)!$ has $2014$ trailing zeros. What is $n$?
4
votes
0answers
70 views

Growth of $n!!\dots !$

The asymptotic growth of the factorial function $n!$ is famously given by Stirling's formula as $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n$$ Is there a similar formula for the iterated ...
5
votes
4answers
362 views

Easy Double Sums Question: $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(m+n)!}$

How to calculate $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \dfrac{1}{(m+n)!} $ ? I don't know how to approach it . Please help :) P.S.I am new to Double Sums and am not able to find any good sources ...
1
vote
4answers
52 views

Does the inequality $ n! > A \cdot B^{2n+1}$ hold for sufficiently large $n$?

Suppose $A,B >0$ are given constants. Is it possible to find a large enough $n \in \mathbb{N}$ such that $$ n! > A \cdot B^{2n+1}?$$
2
votes
1answer
27 views

Is there any way to extend the domain of this function through analytic continuation?

$\prod_{k=2}^x \log k=F(x)$ It looks a lot like the gamma function (a sort of logarithmic factorial), and I wonder if it can be similarly expressed as an integral or something. Any ideas?
0
votes
4answers
33 views

Show $(2n+2)!\geq(n+2)(n+2)!$, $\forall n\in\mathbb{N}$.

It's the last step in a proof, and I just need to show that $$(2n+2)!\geq(n+2)(n+2)!$$ $\forall n\in\mathbb{N}$. I can't seem to do it though, any thoughts?
2
votes
2answers
45 views

Proof of factorial inequality concerning fractions

I'm having trouble with a proof, with the case $n>2$. THEOREM: For every natural number $n∈N$ where $n≠2$, $∑_{i=1}^ni≤n!$ Let us simplify the statement. ...
3
votes
4answers
109 views

Determine if this series $\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$ converges or diverges, and prove your answer?

Determine if this series $$\sum\limits_{n=1}^\infty \frac{(n!)^2}{(2n)!}$$ converges or diverges, and prove your answer? I've been able to prove similar problems, but I'm confused now that there's a ...
0
votes
1answer
48 views

Calculating p-adic valuation $v_p(n)$, using basic properties

Calculating p-adic valuation $v_p(n)$ I'm not confident with the properties of $v_p(n)$ Where $v_p(n) = $ the biggest integer $e$ such that $p^e$ divides $n$, if $n\not=0$, and $+\infty$ if $n=0$. ...
2
votes
3answers
45 views

Lower bound for the falling factorial $(2n)_{n}$

I'm looking for a lower bound for the falling factorial $$(2n)_{n}:= \frac{(2n)!}{n!}$$ I saw on Wikipedia that $n! > \sqrt{2{\pi}n}(\frac{n}{e})^n$ . So I assume that a possible lower bound ...
0
votes
2answers
73 views

Find the number of trailing zeroes. [duplicate]

Find the number of trailing zeroes. $k=1^1\times 2^2\times 3^3\times \cdots \times100^{100}$ It usually involves calculating number of $5$'s in $5^5\times 10^{10}\times 15^{15}\times \cdots\times ...
1
vote
2answers
78 views

Formula for factorial?

I need an equation that defines factorial without using factorial, that also works for $0$. I have seen factorial defined like this: $$n! = 1\cdot2\cdot3\cdot4\cdots n$$ But if we plug $0$ into that, ...
1
vote
2answers
264 views

Why is it defined that $(-1)!!=1$?

Why is it defined that $(-1)!!$ equal to $1$, where $!!$ is the double factorial? I've only seen it defined that $(-1)!!=1$, but I don't see why it should be so.
0
votes
2answers
15 views

What's the difference between derangements and partial derangements?

What's the difference between derangements and partial derangements? I know that Derangements are essentially subfactorials; could anyone explain the difference? I came across this in some local ...
2
votes
1answer
42 views

$\frac{N!}{(N-n)!}$ when $n<<N$

I need to show that for $n<<N$ then $\frac{N!}{(N-n)!} \approx N^{n} $ I can see that $\frac{N!}{(N-n)!} = (N)(N-1)...(N-(n-1))$ and intuitively its clear but I am unable to show rigorously. ...
1
vote
2answers
27 views

Combination vs permutation

A teacher has $5$ books to distribute to some of $20$ children in her class. How many ways are there for her to distribute the books if the books are all the same and no child gets more than one? ...
4
votes
1answer
94 views

Finding missing digits in factorials

14!=871a82b1200 without working out 14!, find a and b I think it has something to do with maths rules regarding 9 or 3 (the digits adding up to either of those numbers) but not entirely sure!
10
votes
2answers
511 views

when is $n!+10$ a perfect square?

When is $n!+10$ is a perfect square ? I have tried and found that only for $n=3$ is $n!+10$ a perfect square. Is there any other solution to this?
2
votes
0answers
231 views

Evaluating $\int e^{\Gamma(x)} dx $ and $\int \pi^{\Gamma(x)} dx $

I don't know how to solve these integrals: $$I_1 =\int e^{\Gamma(x)} dx $$ $$I_2 =\int \pi^{\Gamma(x)} dx $$ As a tenth grader I have no idea what the solutions could be. How would one go about ...
1
vote
1answer
32 views

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$

Find all positive integers $a$, $b$, and $c$ for which $a \choose b$ $b \choose c$ = 2$a \choose c$. Using the theorem ${n! \over k!(n-k)!} = {n \choose k}$ I simplified this down to $(a-c)! = ...
1
vote
1answer
34 views

Divisibility problem using Wilson's theorem: $4(p-3)! + 2$ is divisible by $p$

Prove that $4(p-3)! + 2$ is divisible by $p$, where $p$ is an odd prime. Use Wilson's theorem. I am having trouble trying to bring it in the form where Wilson's theorem can be applied. Any help ...
0
votes
0answers
117 views

Equation with Sum of Factorial and Subfactorial

I am interested in finding solutions to the following equation $$x! + !x = a^3$$ where $x$ and $a$ are natural numbers and $!x$ is the subfactorial of $x$. I've found the solutions $x=1$ and $x=3$. ...
1
vote
0answers
89 views

How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... ...
1
vote
2answers
78 views

Partitioning positive divisors of 100!

Is it possible to partition all positive divisors of 100! (including 1 and 100!) into 2 subsets so that each subset has the same number of integers and the product of all the divisors making up the ...
0
votes
0answers
57 views

Permutations Without Repetitions

Given the set [A,B,C,D] how many distinct ways can I order all four of the members of the set? I see distinct, as a unique set, therefore [A,B,C,D] and [D,C,B,A] ...
5
votes
6answers
168 views

The value of $ \int _{0}^{1}x^{99}(1-x)^{100}dx $ is

The value of $\int _{0}^{1}x^{99}(1-x)^{100}dx $ is Not able to do. I'm trying substituton. But clear failure. Please help.
0
votes
3answers
337 views

Gamma function proof of gamma $\;Γ(1/2) = \sqrt \pi\;$

So our teacher doesnt use the same demonstration as most other sites use for proving that gamma of a half is the square root of pi. I dont understand the demonstration from the first step because he ...