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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26
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 ...
0
votes
0answers
29 views

Modular arithmetic, specifically n! mod m

Is there a theorem that makes solving $$ n! \equiv x \mod m $$ knowing that both $n$ and $m$ are prime? And if not, what would be the best way to go about finding $x$? cheers
1
vote
3answers
71 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
2
votes
3answers
171 views

Prove properties of the factorial (gamma function)

I want to prove the equation is satisfied. 'p' is a natural number. $$\sum_{n=1}^{∞} \frac{\Gamma(n)}{\Gamma(n+p+1)}=\frac{1}{p^2\Gamma(p)}$$ Understandably, This formula can be written as this. ...
1
vote
7answers
183 views

How to show $\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$?

Show that $\,\displaystyle\sum_{i=1}^{n} \binom{i}{2}=\binom{n+1}{3}$. I'm thinking right now (though not getting anywhere with it) that I want to expand out the summation portion to ...
5
votes
2answers
176 views

Finite Summation of Fractional Factorial Series

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

Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ [duplicate]

I'm completely stuck evaluating $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $ how would I go about solving this?
1
vote
1answer
70 views

Evaluate the limit $\lim_{n\to \infty}{\frac{(n+3)!}{n^n}}$

Evaluate the limit $$\lim_{n\to \infty}{\frac{(n+3)!}{n^n}}, n\in \mathbb N$$ I know that the limit is $0$ but how to prove it?
2
votes
5answers
91 views

To evaluate $\lim_{n\to\infty} \dfrac{10^n}{n!} .$

I am having a lot of trouble evaluating $$\lim_{n\to\infty} \dfrac{10^n}{n!} .$$ I know intuitively that $n!$ is much larger and this expression should go to zero but I just can't figure out how to ...
1
vote
3answers
68 views

Calculate $\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$

Could I please have a hint for finding the following limit?$$\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$$
6
votes
8answers
449 views

How to prove that $\lim\limits_{n\to\infty} \frac{n!}{n^2}$ diverges to infinity?

$\lim\limits_{n\to\infty} \dfrac{n!}{n^2} \rightarrow \lim\limits_{n\to\infty}\dfrac{\left(n-1\right)!}{n}$ I can understand that this will go to infinity because the numerator grows faster. I am ...
0
votes
3answers
106 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.
0
votes
2answers
37 views

Trying to determine the number of possible combinations for a password

OVERVIEW: Making a secure password. People tend to use dictionary words as a basis for their passwords. People tend to make minor substitutions on their passwords (password -> p@$$w0rd) Assuming ...
2
votes
2answers
117 views

What is the triple factorial of a negative number, e.g., $-2$?

The triple factorial of a positive integer is computed as $7!!! = 7\cdot 4\cdot 1$. I'm interested in the value of $$(-2)!!!$$ I tried to find this value by using the Wolfram, but I found the ...
-3
votes
1answer
53 views
4
votes
4answers
139 views

Simplify the expression (combination and factorial)

Simplify the following expression: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!}$ My attempt: $\binom{n+1}{3} * \frac{(n-1)! + (n-2)!}{(n+1)!} = \frac{(n+1)!}{3!(n+1-3)!} * \frac{(n-1)! + ...
0
votes
1answer
45 views

number of ways to choose pairs of nonadjacent people from $2k$ people sitting in a circle

The following is problem 19 in Chapter 2 from Richard Stanley's Enumerative Combinatorics, vol. 1 (2nd ed.): Suppose that $2k$ persons are sitting in a circle. In how many ways can they form pairs if ...
-1
votes
1answer
71 views

How to simplify $(a\times b)!$ [closed]

I'm looking for a formula to simplify $(a\times b)!$ just as you can simplify $a^{b+c}$ to $a^b \times a^c$. How can I do this?
0
votes
0answers
11 views

Generalized superfactorial notation.

I would like to know if there's a general shorthand notation for denoting the following product: $$\mathcal{P}=a!\times (a-k)!\times (a-2k)!\times\cdots\times (a-nk)!$$ where $a$ and $k$ both ...
1
vote
1answer
28 views

function of input size N, combination problem [duplicate]

Can someone please elaborate how from $(N+1)+N+(N-1)+(N-2)$ one can get $= 1/2(N+1)(N+2)$? also how to prove that: $(N-1)+(N-2)+...+3+2+1+0 = \frac{N(N-1)}{2} = {N \choose 2}$ ? Thank you!
1
vote
0answers
35 views

Elementary proof about nth differences of nth powers of integer

In a post on Math.SE., a proof sketch was proposed for the proposition below: The sequence of $n$th differences of the sequence of $n$th powers of positive integers, is the constat sequence $n!$. ...
3
votes
2answers
62 views

what is remainder when $(((3!)^{5!})^{7!})^{9!…}$ is divided by 11

$$(((3!)^{5!})^{7!})^{9!...}$$ when divided by 11 what will be the reminder? Hint is appreciated Sorry I do not know how to start this problem, so I have not shown my efforts!
0
votes
0answers
44 views

How to Find Number of Combinations

Here is the problem: In an experiment with eight trials involving the births of three children, what is the theoretical probability that you will get the distribution of: 0 girls-once 1 girl-three ...
3
votes
1answer
24 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
41
votes
0answers
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 ...
27
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) ...
1
vote
2answers
100 views

$2\times 5 \times 8 \ldots \times (3n-1)=?$

Does anybody know if there is a closed form expression using factorials for the above product? I'm not seeing it but I feel like there must be. The recursive relationship corresponding to this ...
1
vote
2answers
45 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
1
vote
1answer
26 views

Intuitive explanation for this Gamma function identity

Wolfram Alpha says that this result is true: $$\frac{\Gamma(n+1)}{\Gamma(\frac{n}{2}+1)} = \frac{\Gamma(\frac{n}{2} +\frac{1}{2})}{\Gamma(\frac{1}{2})} \times 2^n$$ This implies a curious result for ...
1
vote
2answers
70 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, ...
0
votes
1answer
17 views

Combining an outcome of a score

Hey I was wondering how many are the possibilities of combining the scored points of a result such $133:75$ from a basketball game? Considering that there are fouls($1$ point), normal($2$ points) and ...
-6
votes
1answer
38 views

What's the value of n! -1 ?? [closed]

I was practicing mathematics for my studies Then I encountered this problem Evaluate n! - 1 Edit: The problem in the book said exactly Prove: n! -1 = 1!1 + 2!2 + 3!3 ...... n!n This is all ...
3
votes
6answers
64 views

Prove that $2^n(n!)^2 \leq (2n)!$

Prove that $2^n(n!)^2 \leq (2n)!$ One can also use the following result to prove the above: $2 · 6 · 10 · 14 · · · · · (4n − 2) = \frac{(2n)!}{ n!}$. The above relation gives, $(2n)!=2^n n! ...
11
votes
3answers
98 views

Evaluating the factorial-related limit $\lim_{x \to \infty} (x + 1)!^{1 / (x + 1)} - x!^{1/x}$

I'm looking for the limit $$\lim_{x \to \infty} \left[[(x+1)!]^\frac{1}{1+x} - (x!)^\frac{1}{x}\right].$$ I've put the above in a computer program, and evaluated it at very high values of $x$ ...
2
votes
2answers
26 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
1
vote
1answer
26 views

Calculate position in password search

I'm running a password cracker on my own password and I'm trying to calculate how long it will take. I know the rate the software is checking at and I also know the password. The password is $14$ ...
5
votes
2answers
117 views

Find the value of $\,\, \lim_{n \to \infty}\Big(\!\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\!\Big)^{\!1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
3
votes
7answers
157 views

Hint in Proving that $n^2\le n!$ [duplicate]

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
2answers
47 views

Factorial Proof Problem

Suppose $m$ and $n$ are positive integers Prove $m!n! \lt (m+n)!$ I have something along the lines of: Since $1 \lt m+1$ and $2 \lt m+2$ etc.. then: $$n \lt m + n$$ So: $$n! \lt (m+n)!$$ I'm ...
1
vote
2answers
91 views

How many zeroes are there at the end of $36!^{36!}$?

Could you please tell me how many zeroes are there at the end of $36!$ to the power $36!$, i.e., $36!^{36!}$? I have been trying to find out. Read some reviews and answers related this but didn't ...
14
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 $\log {m \choose n}$?
5
votes
1answer
150 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} ...
6
votes
3answers
395 views

How do you find the factorial of a decimal or negative number and what does it show us?

I know that you can find the factorial of positive integers where n!= n(n-1)...2 x 1. However, what if you want to find the factorial of a negative integer or a decimal? I tried to do it on my ...
6
votes
2answers
473 views

Is this a meaningful approach to primes?

Motivation: I tend to be good at recognising patterns and I saw one with factorial: $$ n! = \prod_{i=1} p_i^{J(n,p_i)} $$ where $p_i$ is the $i$'th prime and $$ J(n,i)= \sum_{S=1}^\infty [n/i^S] $$ ...
1
vote
2answers
531 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\\ ...
2
votes
1answer
387 views

How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10} $$ Can ...
5
votes
4answers
92 views

Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically

I am working through the exercises and have spent half a day on one problem so I decided to get some help because I can't figure it out. Show that if $n$ is a positive integer at most equal to $m$, ...
1
vote
2answers
64 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
0
votes
1answer
28 views

Induction Mathematics and Factorials

\usepackage{amsmath} Evaluate the sum $\sum_{k=1}^{n} {k\over (k+1)!}$ $\sum_{k=1}^{1} {1\over (1+1)!} = {1\over 2}$ $\sum_{k=1}^{2} {2\over (2+1)!} = {5\over 6}$ $\sum_{k=1}^{3} {3\over (3+1)!} ...