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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2
votes
1answer
49 views

Proving $\lim((n!)(\frac{e}{n})^n) = \infty$ using elementary method

Is there some easy way to show that $\lim_{n\to\infty} ((n!)(\frac{e}{n})^n) = \infty$ as $n \to \infty$? It looks like Wolfram alpha is using some kind of expansion method to numerically compute this?...
0
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1answer
40 views

solve for the lowest value of $k$ in ${n}\choose {k}$ $\geq$ $x$ , given $x$ and $n$

Let's suppose that $n = 10$ and $x = 500$. I want to find the smallest value of $k$ for which ${n}\choose {k}$ $\geq$ $x$ I can check for all values of $k$ from $1$ to $n/2$ and pick the least one ...
-2
votes
2answers
33 views

how do I do I solve this equation [on hold]

(n+1)! = 110(n-1)! I have searched online for a week now and my textbook does not explain how it got the answer: n=10.
0
votes
0answers
48 views

Factorials in Base 12 | Number Theory

How many zeroes does $66!$ end in when written in base 12? I know that this is the formula we're supposed to use here, but I'm confused how. Isn't this the prime factorization of the factorial? $\...
0
votes
2answers
77 views

Query regarding a theorem in Number Theory

Please refer to Theorem $8$ in the attached picture. I do not understand the significance of the condition $(n+1)!+k>k$. Is it not always true since the inequation implies $(n+1)!>0$ which is ...
1
vote
1answer
50 views

Will someone please explain this equation from the picture

I am trying to understand the following equation: $$\sum_{k=0}^{\infty}\frac{a^k}{k!}\sum_{m=0}^{\infty}\frac{b^m}{m!} = \sum_{n=0}^{\infty}\frac{1}{n!}\sum_{k=0}^{n}\frac{n!}{k!(n-k)!}a^kb^{n-k} = \...
1
vote
3answers
107 views

On the determinant of a Toeplitz-Hessenberg matrix

I am having trouble proving that $$\det \begin{pmatrix} \dfrac{1}{1!} & 1 & 0 & 0 & \cdots & 0 \\ \dfrac{1}{2!} & \dfrac{1}{1!} & 1 & 0 & \cdots & 0 \\ \...
1
vote
1answer
19 views

Relation Gamma function and products

I have been thinking about a specific problem for quite some time. Imagine we have the following product, where $x \in \mathbb{N}$ and $a \in \mathbb{Z^+}$ then we now that the following holds: \begin{...
-6
votes
1answer
44 views

Solving factorials! [closed]

If a!(n-a)!=b!(n-b)!, where a, b, n are naturals. Show that a=b or n=a+b. Also, n>a&b. I needed this prop. to prove one theorem. Help me! I don't need any rigorous proof. I will accept the answer, ...
0
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3answers
59 views

Limit of the sum of a factorial series

Is there any function $f$ over the positive integers such that $$\lim_{n\rightarrow\infty}\frac{\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}}{f(n)} = 1$$ and $$f(n)\not\equiv\sum_{i=0}^{n} {\frac{n!}{(n-i)!}}...
21
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1answer
389 views

Computational results for the sequence $n!+{p_n}!+1$ are, well, very very unusual

Peter and I were discussing in a chat room and I thought that it would be nice to test the sequence $$n!+{p_n}!+1$$ for primality. Then I wrote Peter that I expect much of primes in this sequence, ...
0
votes
2answers
41 views

Designate set $ \lbrace n \in N : 2^{n-1} | n! \rbrace $

Actually I found out that if $n = 2^k$, $k\in Z_{+}$ we can say that it's true. Here's the proof: Let $n = 2^k$, $k\in Z_{+}$ Then $\nu _{2}(n!) = \nu _{2}((2^k)!) = \lfloor \frac{2^k}{2} \rfloor + \...
2
votes
3answers
60 views

Cannot find a calculator that can handle 220! x 40000

For a game I am playing, I am trying to calculate the cost of purchasing all the expansion area squares on a level. the first square costs 40,000, each next square costs an additional 40,000 more than ...
4
votes
3answers
884 views

What algorithms exist to quickly compute the inverse factorial?

I'm interested in algorithms to quickly compute the inverse factorial. I've noted that large factorials have a unique number of digits. How can I use this fact to quickly compute the factorial? Is ...
1
vote
1answer
47 views

What is the smallest integer $k$, for fixed $n$ (or vise versa), does $2^k! \geq 2^{n-k}$?

What is the smallest integer $k$, for fixed $n$ (or vise versa), does $2^k! \geq 2^{n-k}$? I tried to find $k$ for small $n$ so I could get some OEIS hit, but both sides just grow too fast for me to ...
2
votes
1answer
65 views

Functions whose Limit is the Factorial Function

I want to know examples of functions $f(n)$ whose limit is $n!$ Now, when I say "limit", I don't mean $$\lim_{n \to \infty}\frac{f(n)}{n!}=1$$ (I already know functions like that). I'm referring to ...
1
vote
1answer
43 views

Help with factorial simplification (n+1)*(n+1)!+(n+1)!-1 [closed]

Could someone show me the steps to get from $$(n+1)(n+1)!+(n+1)!-1$$ to $$(n+1)!\big((n+1)+1\big)-1$$ I'm having trouble understanding the simplification. Thank you!!!
1
vote
3answers
77 views

limit of $\frac{\log(n^n)}{\log((2n)!)}$ as $n$ approaches $\infty$

I'm trying to solve this: limit of $\frac{\log(n^n)}{\log ((2n)!)}$ as $n$ approaches $\infty$ I know $\log(n^n)=n\log(n)$ using logarithms properties but what about $\log((2n)!)$? Idk how to solve ...
0
votes
4answers
289 views

The asymptotic behavior of $n\ln n -n$ [closed]

How do I show that $$\displaystyle\lim_{n\to\infty}\dfrac{n\ln n - n}{\ln n!}=1?$$
2
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0answers
27 views

Proof Involving Factors Of Arbitrarily Large Numbers [duplicate]

For prime $p$, show whether $$\prod_{p \geq 1} p^{\lfloor \frac{x}{p-1} \rfloor} \sim x!$$ as $x$ approaches infinity, and explain. I don’t know that it’s true, but I thought that it followed, if ...
6
votes
3answers
10k views

Find $\lim_{n \to \infty} \sqrt[n]{n!}$.

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
4
votes
0answers
59 views

Primes $p$ which satisfy $p \mid \sum_{i=1}^{p-1} i!$

This question is inspired from @Mathphile's problem: The value$\sum_{i=1}^n i!$ where $n \in \mathbb{N}$, is only semiprime for $n=3,4$ One can easily solve this conjecture by knowing that $9 \mid ...
53
votes
10answers
12k views

Stirling's formula: proof?

Suppose we want to show that $$ n! \sim \sqrt{2 \pi} n^{n+(1/2)}e^{-n}$$ Instead we could show that $$\lim_{n \to \infty} \frac{n!}{n^{n+(1/2)}e^{-n}} = C$$ where $C$ is a constant. Maybe $C = \sqrt{...
3
votes
0answers
65 views

Prove that $f(n)=n^{2007}-n!$ is an Injective Map

If $f: \mathbb{N} \to \mathbb{Z}$ defined as $f(n)=n^{2007}-n!$ Then Prove that it is an Injective function My try: According to the definition of Injective function: If $p,q \in \mathbb{N}$ and ...
0
votes
3answers
60 views

A series involving factorial

Series from $n=0$ to infinity of $n!/1000^n$ I know the limit of $n!$ is infinity and $1000^n$ is also infinity. In this regard, I really don't see how L'Hopital's rule can work in this case. How do ...
2
votes
1answer
51 views

Show that $\frac{(m!)^{1/m}}{m/e}$ is a decreasing function of $m$

Show that $\dfrac{(m!)^{1/m}}{m/e}$ is a decreasing function of $m$. Here is my proof. I would like to see others, preferably simpler. I have shown in Proof explanation $\lim\limits_{n\to\infty}\...
3
votes
2answers
156 views

Which one is bigger $100^{300}$ or $300!$?

How to find which one is bigger $100^{300}$ or $300!$ without using a calculator? I have tried it for whole 2 years but could not find it yet.
14
votes
9answers
6k views

Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$.

I can't seem to find a good way to solve this. I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one. Any ...
1
vote
2answers
79 views

Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
1
vote
1answer
35 views

Expressing the coefficients of $(1-x)^{1/4}$ using factorials

From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$ (1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n. $$ However, can I do the same ...
2
votes
2answers
69 views

can we perform modulo operator on a fraction on both of it's numerator and denominator?

I want to calculate nCr (mod $10^9+1)$.so for calculating nCr we have: $$nCr=\frac{n!}{r!(n-r)!}$$ so I want to know whether it is true that I perform modulo operator to numerator and denominator ...
-1
votes
2answers
36 views

Show that the sequence is not bounded above

I must show that the sequence is not bounded above: $a_n =\frac{n^n}{n!}$, I tried to use proof by contradiction: suppose there is some $k$ such that $a_n\le k$, then $n^n \le kn!$, $n*n*n...*n \le ...
2
votes
1answer
58 views

How to calculate a complicated permutation?

I'm writing a play that features a lot of randomization, which will mean that it is different every time, and I'm trying to calculate the number of performance possibilities. It features a number of ...
0
votes
1answer
71 views

Beta function and gamma function

I would like to ask if someone could help me with following equation. \begin{equation} \Gamma(m)\,\Gamma(n) = \int_{0}^{\infty}x^{m-1}e^{-x}\,dx\,\int_{0}^{\infty}y^{n-1}e^{-y}\,dy \end{equation} \...
0
votes
0answers
48 views

How can I simplify $\prod\limits_{k=1}^{n}3k+1$?

I've found that I can simplify $\prod\limits_{k=1}^{n}2k-1$ to $\frac{(2\cdot n)!}{2^n\cdot n!}$, for example: $\prod\limits_{k=1}^{4}2k-1=1\cdot3\cdot5\cdot7=\frac{1\cdot2\cdot3\cdot4\cdot5\cdot6\...
1
vote
1answer
31 views

Minimum of maximum with factorial

Let $f:\mathbb N_+\to\mathbb N_+, f(n)=\min\{\max\{k!,(n-k)!,(n!-k!(n-k)!)\}|k\in\mathbb N_+, n-k> 0,\ n!-k!(n-k)!> 0 \}$. What's the asypmtotic growth of $f(n)$? Is it true that $f(n)=\Theta((...
0
votes
1answer
48 views

For which $n$ is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$?

For how many values of $n$, is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$? Further more, is there a way to ...
5
votes
4answers
342 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 ...
89
votes
7answers
8k 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.
2
votes
2answers
264 views

An inequality with exponents, factorials and nth roots!

Problem: Prove for natural numbers $n > 2$, $$(\sqrt{2!}-1)((3!)^{\frac{1}{3}}-\sqrt{2!})\cdots(((n+1)!)^{\frac{1}{n+1}}-(n!)^{\frac{1}{n}}) < \frac{n!}{(n+1)^n}.$$ I am unable to do this one. ...
1
vote
1answer
62 views

Wallis integral and gamma function

I would like to ask if anyone would help me to explain how to reach the following relation. \begin{equation} \int_0^1 \left( 1-x^{\frac{1}{p}} \right)^q dx= \frac{p!\,q!}{(p+q)!} \end{equation} If we ...
4
votes
2answers
127 views

How to calculate limit as $n$ tends to infinity of $\frac{(n+1)^{n^2+n+1}}{n! (n+2)^{n^2+1}}$?

This question stems from and old revision of this question, in which an upper bound for $n!$ was asked for. The original bound was incorrect. In fact, I want to show that the given expression ...
3
votes
2answers
109 views

Show that $n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ for $n\ge 2$

$n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ For very small values of $n$ (i.e. $2\le n\le 6$) the function on the right nicely approximates $n!$ before significantly overtaking it. I don't have ...
1
vote
0answers
31 views

Factorial for positive fractions

I know that factorial is also represented by Gamma function and we have exact value of factorial of (1/2), can someone help me how to find numerically(approximate) the factorial of positive fractions ...
6
votes
7answers
2k views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any ...
3
votes
1answer
78 views

Proof by induction for nth derivative

Show the following hold by induction: $$\frac {d^n}{dx^n}\frac {e^x - 1}{x} = (-1)^n \frac{n!}{x^{n+1}} \left( e^x \left(\sum_{k=0}^{n} (-1)^{k} \frac{x^k}{k!}\right) - 1 \right)$$ Proof. It's not ...
1
vote
2answers
42 views

Simplifying factorial with different coefficient [closed]

$$\frac{(pn)!}{(qn)!},\quad p\not = q$$ If possible, how could I simplify the above factorial?
3
votes
2answers
141 views

Does the sum of reciprocals of primorials converge?

It is well known that the sum $$ \sum _{{k=0}}^{\infty }{\frac {x^{k}}{k!}} $$ converges to $e^{x}$. In particular, for $x=1$ we have $\sum _{{k=0}}^{\infty }{\frac {1}{k!}}=e$. But what about the ...
5
votes
3answers
101 views

Why do the factorials appear in differences of consecutive powers?

Why do the factorials appear when repeatedly taking the differences of consecutive powers? Or rather why is the $n_{th}$ factorial equal to the $n_{th}$ difference of $(k+1)^{n}-k^n$? I'm having ...
1
vote
2answers
43 views

Logarithm of factorial equal to sum of logarithm of primes

Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that \begin{align} \log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...