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

9
votes
1answer
185 views

One-Line Proof for $n! \geq (\frac n e)^n$

I was told to find a one-line proof for $n! \geq (\frac n e)^n$. I'm advised that Stirling's formula is not helpful. I've spent a little bit of time on it, but the solution is not coming to me. I feel ...
6
votes
1answer
59 views

maximize a function which contains factorials

Suppose I have a function $$ f(k) = \binom{500}{k} \binom{500}{1100-3k}$$ where $k$ is an integer from $200$ to $366$. How can I find the maximum analytically?
11
votes
3answers
733 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...
2
votes
2answers
84 views

How can I determine convergence of the series of $\frac {(2n)!}{2^{2n}(n!)^2}$

Does the series $$\sum_{n = 1} ^ {\infty} \frac {(2n)!}{2^{2n}(n!)^2}$$ converge? The ratio test doesn't work for the series.
0
votes
2answers
100 views

An identity for the product of even numbers (double factorial)

I'm unable to prove this identity: Prove that: $2\cdot 4 \cdot 6 \cdot 8 \cdots 2n = 2^n \cdot n!$ Wouldnt it be like this? $ 2(1 \cdot 2\cdot 3\cdot 4 \cdots n)= 2 \cdot n!$
6
votes
2answers
561 views

How can I unfactorilize number?

Consider the equation $x! = y$ Say we know $y$ and were trying to find $x$: What method could I use to get $x$ (e.g. a closed formula)?
6
votes
4answers
213 views

Direct proof that $n!$ divides $(n+1)(n+2)\cdots(2n)$

I've recently run across a false direct proof that $n!$ divides $(n+1)(n+2)\cdots (2n)$ here on math.stackexchange. The proof is here prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer (it ...
0
votes
2answers
100 views

How does $n!^2$ divide $(2n)!$? [duplicate]

How can I show that $(n!)^2$ divides $(2n)!$, where $n$ is a natural number? So far I've noticed that we can rewrite $\dfrac{(2n)!}{(n)!^2}$ as a combination and we know that combinations are always ...
0
votes
5answers
69 views

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$

Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$ I've already considered using l'Hoptials rules but I cannot take the derivative of a factorial (as it is a discrete function). Thanks
8
votes
3answers
990 views

What remainder does 34! leave when divided by 71 ??

What is the remainder of $ \frac{34!}{71} $? Is there an objective way of solving this? I came across a solution which straight away starts by stating that $69!$ mod $71$ equals $1$ and I lost it ...
2
votes
2answers
340 views

Is there a name for $\frac{n!}{m!}$?

Is there a name or short way of writing of $\frac{n!}{m!}$? I've searched and the closest I could find was binomial coefficient. Is there any other way?
3
votes
9answers
270 views

How can one prove $\lim \frac{1}{(n!)^{\frac 1 n}} = 0$?

I have tried bounding the terms by $\dfrac 1 {2^{\frac 1 n}}$, but this clearly cannot be made as small as possible.
1
vote
1answer
46 views

Is this equation for $2^k!$ correct?

I couldn't find any equation for $2^k!$ so I came up with an equation that appears to work for the factorial of a power of $2$. However, I'm having problems proving it. My equation: $$ \def\x{\times} ...
1
vote
3answers
92 views

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

So guys, how can I evaluate and prove that $\lim_{n \to\infty} \to \frac{(2n-1)!!}{(2n)!!}=0$. Any ideas are welcomed.
5
votes
1answer
101 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are ...
0
votes
1answer
57 views

Smallest value of n such that the product $n!$ ends in at least 10 zeros.

What is the smallest value of $n$ such that the product $n!$ ends in at least 10 zeros? I tried to do this by multiplying each number but it didn't work. Please help.
4
votes
6answers
131 views

Product of r consecutive integers is divisible by r!

Well in a book i am reading it is given that you can also prove this by showing that Every prime factor is contained in $(n+r)!$ as often at least as it is contained in $n!r!$. How does this prove ...
1
vote
0answers
9 views

Simplifcation of the Function Required

I want to simplify the following expression $$ \sum_{x=0}^{D+m} \frac {x!} {(n+1)_x} (\frac {\theta_{eff}} {\mu})^x {D+m \choose x} $$ The parameters $D, m, n, \theta_{eff}, \mu, $ are constants. ...
6
votes
4answers
2k views

Factorial of 1,e+80

Recently I started being very fascinated in logistics, and out of the blue came the question into my head, what is the factorial of the amount of atoms in the observeable universe, which is said to be ...
3
votes
3answers
123 views

The sum $1!+2!+3!+…+2007!$ is not a perfect square

Today my teacher told me to prove this:- Prove that $1!+2!+3!+...+2007!=\sum_{n=1}^{2007}(n!) $ is neither a perfect square nor a perfect cube. Not getting any idea. Please help. Thanks in advance.
0
votes
4answers
636 views

Mathematical Proof of zero factorial is 1 . [duplicate]

I wanted to know is there any mathematical proof underlying 0! = 1 rather than qualitative analysis
1
vote
0answers
48 views

The Number of The 0's in a Factorial

I need to find that the number of the 0's at the end of the number is odd or even in a factorial. For example: $0! = 1$ (Even) $5! = 120 $ (Odd) $18! = 6402373705728000 $ (Odd) Dou you have any ...
11
votes
3answers
157 views

$1!+2!+\ldots+n!$ cannot be the square of a positive integer

I have to prove that $1!+2!+\ldots+n!$ cannot be the square of a positive integer, $\forall n\geq4$. I've tried to do this with induction, but I don't seem to reach any satisfactory conclusion. Any ...
0
votes
4answers
52 views

$(n+1)!>n^2, \forall n\ge 4$

The base case is clear, since if $n\gt 4, 5!=120\gt 25=5^2$ So assume $n=k$ which shows that $k!\gt k^2$. Then if $n=k+1$, $$(k+1)!=(k+1)k!$$ $$\gt (k+1)k^2 $$ Induction argument $$=k^3+k^2$$ $$\gt ...
-1
votes
3answers
120 views

How to interpret $(2n)!$

It's all in question: how to interpret the factorial from $2n$? Is $(2n)!$ equal to $n!\times n!$ ? The problem is in Combinations if the combinations is $\binom{2n}3$. P.S. The main problem is ...
2
votes
1answer
35 views

Definition of factorial function

Is there a definition of the factorial function (for natural numbers) in the language of ordered rings?
1
vote
0answers
66 views

Digits of $n$ factorial

With the notable exception of $0$, for large enough $n$, the digits in base $10$ for $n!$ seem pretty much uniformly distributed (I have also checked for other few bases $> 2$). Have anyone ...
2
votes
5answers
117 views

Hint in Proving that $n^2\le n!$

The problem is to find the values of n such $n^2\le n!$. I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$} I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...
0
votes
1answer
28 views

AM I doing this right? - How many binary words of length 8 are there that contain at least six 1's?

How many binary words of length 8 are there that contain at least six 1's? This is what I have: 8!/6!2! = 28 words Is this the correct answer?
2
votes
3answers
794 views

Simplify the expression $(2n)!/(2n+2)!$

I'm a little confused as to how $(2n)!/(2n+2)!$ looks when written out. Basically I'm trying to visualise it so that I know how to cancel this and like terms in future.
0
votes
0answers
61 views

lower bound for factorials product

For $n$ positive integer, let $F(n) = 1! × 2! × 3! × 4! × \cdots × n!$, product of factorial(i) for $i$ in $[1\ldots n]$. Let $G(n) = \{i \in [1\ldots n],\text{ such that }n\mid F(i)\}$. It is ...
1
vote
1answer
346 views

Factorials in Sigma Notation

Geometric Series one would use $S_n = \dfrac{a_1\cdot (1 - r^n)}{(1 -r)}$. Arithmetic Series one would use $S_n = \dfrac{n\cdot (a_1 + a_n)}{2}$. But how would I convert a sigma notation problem with ...
7
votes
2answers
68 views

The number of primes in the factorization of $N!$

Is there an approximation to the number of primes in the factorization of $N!$? For example: For $N=10$, this number is $15$. For $N=100$, this number is $239$. For $N=1000$, this number is $2877$. ...
1
vote
1answer
53 views

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$.

Prove or disprove: $(\frac{1}{n})^n(1 - \frac{1}{n})^{n^2-n} \simeq \frac{1}{n!}$ as $n \rightarrow \infty$. I'm trying to prove the statement by building on my observation that $(1-\frac{1}{n})^n$ ...
0
votes
2answers
30 views

Factorial simplication question

How does the following: $$(k+1)! - 1 + (k+1).(k+1)!$$ simplify to: $$ (1+k+1).(k+1)! - 1 $$ and then $$(k+2)! - 1$$ I just can't seem to see how that works, I've tried writing out the factorials ...
0
votes
2answers
29 views

How can we prove this = 1 for all n

$\displaystyle n!-\sum_{k=1}^{n-1}k\cdot k!$ By computing this by hand for several small values of $n$ I can see that it is always equal to 1. But I can't see how to prove that.
1
vote
3answers
79 views

Factoring added factorials

How do I facilitate prime factorization without brute-forcing the 600+ digit number? For example, how would I factor (82! + 83! + 84!) ?
1
vote
1answer
72 views

Upper bound for $n!$

Let $a\in\mathbb{N}$. is there an upper bound be for the smallest n so that $n!>a$? It doesn't have to be a good upper bound, just something that works. Thanks.
1
vote
2answers
70 views

Number of primes in $[30! + 2, 30! + 30]$

How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$, where $n!$ is defined as: $$n!= n(n-1)(n-2)\cdots3\times2\times1$$ Using Fermat's Theorem: $130=1\mod31$, (since $31 \in ...
1
vote
4answers
99 views

Demonstrate that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} \simeq 1.7 \sqrt{n}$

As in the title, I know that $\displaystyle \frac{(2n - 2)!!}{(2n - 3)!!} = \frac{(2n - 2)(2n - 4)\cdots 4 \cdot 2}{(2n - 3)(2n - 5) \cdots 3 \cdot 1} \simeq 1.7 \sqrt{n}$ Could you give some hint ...
0
votes
3answers
54 views

series calculation involving factorial

How would one calculate following $$\sum_{k=2}^\infty \frac{k^2+3k}{k!}$$ I searched youtube for tutorials (patricJMT and other sources) where I usually find answers for my math problems, I think I ...
2
votes
1answer
55 views

Calculate sum wtih binomial coefficients

I need help with finding the sum of $\sum \limits_{k=0}^{n} \frac{1}{k+1}{n\choose k}x^{k+1}$
0
votes
2answers
49 views

How to calculate this sum

How do you calculate this sum $ \sum \limits_{k=1}^{n} \frac{k}{n^k}{n\choose k}$ ?
0
votes
1answer
25 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
9
votes
3answers
157 views

Show that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$

I'd like to prove that $\,\,n!<\mathrm{e}\left(\frac{n}{2}\right)^n$. What I have so far: $\sqrt[n]{n!} = \sqrt[n]{1\cdot 2 \cdot \ldots \cdot n} \leq \frac{1+\ldots ...
1
vote
2answers
109 views

Can non-integer factorials be calculated without numerical integration?

I saw a strange way to write the factorial function somewhere and after some integration by parts, it all sure enough worked out. $$ n! = \int_0^\infty x^{n}e^{-x}dx $$ $$ ...
2
votes
1answer
64 views

Field Theory, Factor Ring, Polynomials

I have the following problems: (1) Let $g=X^2+\overline{4}$ and $h=X^2+\overline{2}$ be polynomials in $(\mathbb{Z}/\mathbb{Z}7)[X]$. $L$ and $K$ are the splitting fields of $g$ and $h$ over ...
2
votes
1answer
57 views

Approximation of a factorial

In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which ...
1
vote
3answers
257 views

Prove that no number in this list is prime - Formatting a proof advice

Question: Let $n \in \mathbb{Z}$ where $n \geq 2$, prove no number in the list: $$n! + 2, n! + 3,...,n! + n$$ is prime. I have written my proof exactly as follows: Proof: $P(n) = n! + n = ...
2
votes
1answer
84 views

Using combinatorial reasoning to show $n!=\binom{n}{0}D_n+\binom{n}{1}D_{n-1}+\dots+\binom{n}{n}D_0$

How can one use combinatorial reasoning to show that $$n!=\dbinom{n}{0}D_n+\dbinom{n}{1}D_{n-1}+\dbinom{n}{2}D_{n-2}+....+\dbinom{n}{n-1}D_1+\dbinom{n}{n}D_0$$ Now $D$ stands for deranged which is a ...