Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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12
votes
3answers
6k views

Proof that $\frac{(2n)!}{2^n}$ is integer

I am trying to prove that $\dfrac{(2n)!}{2^n}$ is integer. So I have tried it by induction, I have took $n=1$, for which we would have $2/2=1$ is integer. So for $n=k$ it is true, so now comes time ...
13
votes
1answer
7k views

Factorial of a non-integer number

My TI-83 calculator doesnt allow me to do this, but using Windows calculator, I can compute the factorial of say 5.8. What does this mean and how does it work?
8
votes
1answer
271 views

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

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 $r-1$...
10
votes
8answers
714 views

Telescoping series of form $\sum (n+1)\cdots(n+k)$ [duplicate]

Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdots(n+k)$ e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + \cdots + n(n+1)(n+2)$ How does it do it? EDIT: We can rewrite as: $\sum {(n+k)!...
9
votes
2answers
152 views

Finding limit of sequence: $\lim _{n \to \infty} {\frac{n!}{n^k(n-k)!}}=1$

$k$ is nonnegative integer. I want to show that$$ \lim _{n \to \infty} {\frac{n!}{n^k(n-k)!}}=1$$ My try : $$ \frac{n!}{n^k(n-k)!} = \frac{n}{n} \frac{n-1}{n} \cdots \frac{n-k+1}{n}$$ I wanted use ...
8
votes
7answers
692 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 $$\lim\limits_{n\to\infty}\frac{\ln(n!)}...
4
votes
2answers
675 views

Proof of Stirling's Formula using Trapezoid rule and Wallis Product

I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with $ \frac {n!}{(n/e)^n \sqrt n}$ $ \rightarrow C$ where $C$ is derived from Wallis product. ...
8
votes
1answer
294 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
4
votes
2answers
131 views

Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$

Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1? Thanks
3
votes
2answers
1k views

Double factorial series

My question is pretty simple. Since $n! \gt n!!$, it's clear by the comparison test that $\sum_{n=0}^\infty \frac {1}{n!!}$ converges. But what value does the sum converge to? How does one go ...
2
votes
1answer
84 views

Is it possible to find $n-1$ consecutive composite integers

Given an integer $n\geq 2$ ,can we always find an integer $m$ such that each of the $n-1$ consecutive integers $m+2,m+3,.....,m+n$ are composite?
2
votes
2answers
91 views

Why $ \lim_{n\rightarrow \infty} \frac{n!}{n^{k}(n-k)! } =1 $?

I was on brilliant.org learning probability. There was a process explaining how the distribution of a Poisson Random Variable can be obtained from a Binomial Random Variable. Consider the binomial ...
1
vote
2answers
102 views

How to show that for all k, $k! \ge (k/2)^{k/2}$

I'm working on a homework problem that has me showing a "$\Omega(n\log k)$ lower bound on the number of comparisons needed to sort a sequence of $n$ elements, when the input sequence consists of $\...
0
votes
6answers
611 views

How to simplify the summation kk! without using induction?

$$\sum_{k=1}^nk(k!)$$ I know the answer is (n+1)!-1..I can solve this question using principle of mathematical induction...but I would like to know if there is any other alternative approach
11
votes
5answers
3k views

Prove that $\gcd(n!+1,(n+1)!+1)=1$

I'd like to solve this one similarly to my previous question: Is this a Valid proof for $(2n+1,3n+1)=1$? I did find a somewhat related post that uses a different method: How to show that $\gcd(n! + 1,...
5
votes
3answers
911 views

Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]

The problem is following, prove that: $$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$ I've tried solving this problem using mathematical induction, but I don'...
3
votes
5answers
515 views

Factorial of 0 - a convenience? [duplicate]

If I am correct in stating that a factorial of a number ( of entities ) is the number of ways in which those entities can be arranged, then my question is as simple as asking - how do you conceive the ...
2
votes
2answers
189 views

Convergence of $a_n= \frac{n!}{n^n}$? [duplicate]

I'm trying to find out whether sequence $a_n= \frac{n!}{n^n}$ converges or not. My textbook says to compare with $\frac{1}{n}$, and my answer sheet says that $\lim_{x\to \infty} {\frac{n!}{n^n}}\leq \...
2
votes
2answers
417 views

Proof for convergence of a given progression $a_n := n^n / n!$

"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable. (a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$ [...]" I am having problems ...
1
vote
3answers
62 views

Proof that $\lim\limits_{h \to \infty} \frac{h!}{h^k(h-k)!}=1 $ for any $ k $ [duplicate]

I kind of barely understand this in some way, and I think I would understand it better by a formal proof. Where do I start?
1
vote
2answers
292 views

Prove by induction that $n! > n^2$ [duplicate]

How does one prove by induction that $n! > n^2$ for $n \geq 4$
175
votes
6answers
9k views

How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
91
votes
2answers
5k views

Prove elementarily that $\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}$ is strictly decreasing

Prove without calculus that the sequence $$L_{n}=\sqrt[n+1] {(n+1)!} - \sqrt[n] {n!}, \space n\in \mathbb N$$ is strictly decreasing.
56
votes
16answers
8k 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 ...
23
votes
2answers
1k views

How to prove this $\pi$ formula? [duplicate]

I am hoping to find out where the formula $$\frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{k!}{\left(2k+1\right)!!}$$ comes from. I can't see how one could begin to prove it.
16
votes
13answers
8k 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.
13
votes
2answers
3k 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.
5
votes
4answers
345 views

Showing $\lim_{n \to +\infty} \log(n!)/(n\log n) = 1$ without using Stirling approximation

As a passage of a bigger limit I have to show that $$ \lim_{ n \to \infty } \frac{\log(n!)}{n\log(n)} = 1. $$ I think it could be done using Stirling approximation, but I'm wondering if there's a way ...
9
votes
2answers
411 views

Another evaluating limit question: $\lim\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}$

How do I begin to evaluate this limit: $$\lim_{n\to \infty}\ \frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}\;?$$ Thanks a lot.
10
votes
4answers
2k views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
4
votes
1answer
390 views

Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

Stirling's approximation of the factorial for even numbers is given by $$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$ Further, the Euler numbers grow quite rapidly for large ...
13
votes
5answers
10k 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 ?
5
votes
8answers
257 views

Why does this series $\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$ converge?

The following series $$\sum_{n=0}^{\infty} \frac{(n!)^{2}}{(2n)!}$$ converges. It fails the divergence test, but once I apply the ratio test, the limit is always equal to $\infty$. Unless you cannot ...
4
votes
2answers
12k views

What does the factorial of a negative number signify?

I understand that the factorial gives the number of arrangements. For example: the factorial of zero i.e. an empty set ( doesn't occur) is 1. As the empty set can be arranged only in 1 way - i.e. by ...
6
votes
2answers
202 views

How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)…(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$

Does anyone have any idea how to prove that $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$$
5
votes
4answers
195 views

A binomial inequality with factorial fractions: $\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+…+\frac{1}{n!}$

Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for $n>1 , n \in \mathbb{N}$.
3
votes
2answers
419 views

Solving Induction $\prod\limits_{i=1}^{n-1}\left(1+\frac{1}{i}\right)^{i} = \frac{n^{n}}{n!}$

I try to solve this by induction: $$ \prod_{i=1}^{n-1}\left(1+\frac{1}{i} \right)^{i} = \frac{n^{n}}{n!} $$ This leads me to: $$ \prod_{i=1}^{n+1-1}\left(1+\frac{1}{i}\right)^{i} = \frac{(n+1)^{n+1}}...
3
votes
5answers
800 views
1
vote
4answers
162 views

Find $\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}$ [duplicate]

I am having trouble showing $$\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}=e.$$
10
votes
4answers
751 views

Does $a!b!$ always divide $(a+b)!$

Hello the question is as stated above and is given to us in the context of group theory, specifically under the heading of isomorphism and products. I would write down what I have tried so far but I ...
4
votes
1answer
8k views

Prove Pascal's Rule Algebraically

I am trying to prove Pascal's Rule algebraically but I'm stuck on simplifying the numerator. This is the last step that I have, but I'm not sure where to go from here $$=\frac{\left[(k-1)(n-k)!+k(n-1-...
4
votes
3answers
396 views

What is $\aleph_0!$?

What is $\aleph_0!$ ? I know that in the original definition the factorial is defined for natural numbers but, what if we extend this concept to cardinal numbers? This concept has been extended to ...
3
votes
3answers
334 views

A Diophantine equation involving factorial

Does the following diophantine equation have no positive integer solutions? $$x^3-y^3=z!$$ Many problems involving diophantine equations are hard. Is it an open problem? I hope someone can give ...
3
votes
1answer
566 views

An integral formula for the reciprocal gamma function

I'm looking to compute an exact integral formula for the reciprocal of the double factorial function, $(2n-1)!!$, or just as easily for the reciprocal gamma function, $\Gamma\left(n+\frac{1}{2}\right)$...
3
votes
2answers
440 views

What is the closed form approximation of the asymptotic growth rate of the superfactorial function?

The asymptotic growth rate of the hyperfactorial function (defined to be: $H(n)=\prod^n_{k=1}k^k$) is apparently (approximately) equal to: I'm curious as to how this result is obtained, and am also ...
3
votes
7answers
287 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 ...
2
votes
2answers
982 views

Limit of a fraction of double factorials

How can we show that $\begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*}$ where $\begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*}$ ...
2
votes
0answers
86 views

Trying to generalize an inequality from Jitsuro Nagura: Does this work?

I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$. In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$: $$\frac{1}{n}\...
0
votes
2answers
224 views

Using the squeeze theorem to determine a limit $\lim_{n\to\infty} (n!)^{\frac{1}{n^2}}$

Currently learning how to use the squeeze theorem to determine a limit. The exercise I'm working on is finding the limit of: $\lim \ (n!)^{\frac{1}{n^2}}$ So far what I have is: $\lim \ (n!)^{\...
10
votes
0answers
176 views

Finding $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$ [duplicate]

How to find : $$\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!} +\frac{5}{3!+4!+5!}+\cdots+\frac{2008}{2006!+2007!+2008!}$$