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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3
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2answers
74 views

How to define a factorial using multiple sets

I am currently studying a photography course, and I have run into a bit of difficulty with one of my projects in relation to combinatorics. I have a key rack and there are 39 hooks on this key rack. I ...
0
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1answer
109 views

Derivative of $\frac{e^x}{x!}$

I am having a bit of trouble putting all the differentiation rules together with the following problem: $$ \frac{d}{da} \Bigg(\frac{a^x}{x!}e^{-a}\Bigg)$$ Where $x$ is a discrete variable and $a$ is ...
1
vote
1answer
330 views

Comparing rates of change: which function increases faster?

I am comparing two functions for $x \ge 1$: $$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$ $$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
1
vote
1answer
54 views

Is this a valid way to evaluate a function based on factorials? What would be a better way?

I am working on the following factorial function: $$f(x) = [\ln(\lfloor\frac{x}{11}\rfloor!) - \ln(\lfloor\frac{x}{12}\rfloor!) - \ln(\lfloor\frac{x}{132}\rfloor!)] + ...
0
votes
1answer
124 views

Proving that a specific gamma function is a guaranteed lower bound for a factorial function

In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that: $$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$ I have ...
1
vote
1answer
58 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
1
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0answers
42 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
3
votes
1answer
697 views

Funny graph of $x!$ by a graphing program

I obtained a bizarre graph of $x!$, a function which I believe is only defined at positive integer domain. What causes such an error?. It would be interesting if someone can explain the method such a ...
8
votes
1answer
237 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 ...
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0answers
143 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
1
vote
1answer
208 views

Understanding the upper and lower bounds of the error estimate in Stirling's Approximation

Based on the Wikipedia article on Stirling Approximation: $n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$ where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$ How would this ...
15
votes
3answers
208 views

Proving that $\frac{(k!)!}{k!^{(k-1)!}}$ is an integer

I have to prove that: $$\frac{(k!)!}{k!^{(k-1)!}} \in \Bbb Z$$ for any $k \geq 1, k \in \Bbb N$ Tried doing $t = k!$ which would give $$\frac{t!}{t^{t/k}}$$ But I think I just made it harder, and ...
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2answers
122 views

How to show the double factorial isn't a polynomial

$(2n-1)!! = \dfrac{(2n)!}{2^{n} \times n!}$ I was wondering how you prove the double factorial is exponential. I guess you have to prove that for all $m$ and $\alpha$ that there exists an $n$ such ...
2
votes
2answers
258 views

Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation

I am trying to get the lower bound for: $f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
2
votes
2answers
57 views

Proving factorial attribute

Given that : $$ \sum_{i=1}^{k} a_i = n $$ I am asked to prove that: $$\prod_{j=1}^{k} a_j! $$ divises $n!$ I saw that it works for $k=1$, and for $k=2$ I tried : $$\frac{n!}{a_1! a_2!} = \frac{(a_1 ...
3
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1answer
220 views

Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?

In Ramanujan's proof of Bertrand's Postulate, Ramanujan states: $\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$ where: $\vartheta(x) = \sum_{p \le x} ...
8
votes
0answers
459 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
2
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3answers
123 views

How to find this expression $1000! \mod 3^{300}$

How to find this expression $(1000!\mod 3^{300})$?
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2answers
314 views

Simplify summation of factorials

Hello I guess this equality is true but I don't know how to solve it. $$\sum_{x=0}^{m(1-\text{sel})} (m-1-x)! (m \cdot \text{sel}) \frac{(m(1-\text{sel}))!}{(m(1-\text{sel})-x)!}(x+1) = ...
4
votes
3answers
1k views

Calculating limit involving factorials.

I want to show that $\lim\limits_{k\to\infty} \frac{\pi^kk!}{(2k+1)!} = 0$. I've been trying to use the squeeze theorem, but am having a hard time finding some expression $P$ involving $k$ that is ...
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11answers
17k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?
2
votes
3answers
63 views

Constrictions on A.P with factorials.

There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression. Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
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3answers
310 views

Compact formula for $\sum_k k!$ [duplicate]

Is there any compact formula for: $$\sum_{k=0}^n k!$$ I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$ I've ...
2
votes
1answer
258 views

How to perform the summation/addition of binomial coefficients?

From my textbook: $$ \begin{align} \sum_{k=0}^n \binom {m+k}m &= \binom {m+n}m + \sum_{k=0}^{n-1} \binom {m+k}m\\\\\\ &= \binom {m+n}m + \binom {m+n}{m+1}\\\\\\ &= \binom {m+1+n}{m+1} ...
4
votes
4answers
8k views

What are the rules for factorial manipulation?

I know that $$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$ thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations ...
4
votes
2answers
168 views

Series involving factorials

How would one go about proving $$\int_{0}^1\frac{e^x-1}{x/2}\ dx=\sum_{n=0}^\infty\frac{1}{\binom{n+2}{2}}\frac{1}{n!}(0!+1!+2!+3!+...+n!)$$
7
votes
4answers
338 views

Combinatorial proof to $n! = (n-1)[(n-1)! + (n-2)!]$

It is for sure true that $n! = (n-1)[(n-1)! + (n-2)!]$ Since: $(n-1)(n-1)! + (n-1)(n-2)! = $ $(n-1)(n-1)! + (n-1)! =$ $ (n-1)!(n-1+1) = (n-1)!n = n! $ Today my friend told me that there is a ...
0
votes
1answer
320 views

Handling summations with two variables

If I have a summation with let's say $x=0 \dots 500$ and $y=0\dots1500$ $500 \choose x$ $ 1500 \choose y$ $\dfrac{1}{2^{500}}\dfrac{2^{1500-y}}{3^{1500}}$, How would I handle the constant? If I ...
2
votes
2answers
297 views

How to prove this inequality with factorials [duplicate]

This is what I am trying to solve: $n!>n^{\frac {n}{2}}$ I tried with induction and somehow it doesn`t work this way, I tried with logarithms and also didn´t find a way. Is there an elementary(if ...
5
votes
1answer
393 views

Question about Ramanujan's proof of Bertrand's Postulate

I am reviewing Ramanujan's proof of Bertrand's Postulate which can be found here. At step #7, he writes: "But it is easy to see that..." $\log\Gamma(x) - 2\log\Gamma\left(\frac{1}{2}x + ...
3
votes
1answer
102 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} ...
3
votes
2answers
226 views

Factorials and Arithmetic Progression.

Are there sets of factorials $(a_1!,a_2!,a_3!,\dots,a_n!)$, such that they exist in Arithmetic progression. $n$ is a natural number I don't see any such examples(Except for $n=2$). And I don't see ...
3
votes
2answers
524 views

What does it mean to “have a multiplicative inverse of modulo 10!”?

Here's the question: What's the smallest integer > 1 that has a multiplicative inverse modulo 10! (that is, 10 factorial)? What does that mean? I understand that: We say that x is the ...
2
votes
3answers
158 views

Showing that $3n<n!$ whenever $n$ is an integer with $n \geq 7$

How can we show that: $$3n< n!$$ whenever $n$ is an integer such that $n \geq 7$ ? I was thinking that we can prove this by showing that such case is true with any integer above 7, but ...
3
votes
3answers
111 views

Evaluate $\sum_{k=1}^{n}(k^2 \cdot (k+1)!)$

We have to evaluate the following: $$1^2 \cdot 2! + 2^2 \cdot 3! + \cdots + n^2 \cdot (n+1)! =\sum_{k=1}^{n} k^2 \cdot (k+1)!$$ Any hints ?
10
votes
1answer
143 views

Can this product be written so that symmetry is manifest?

Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression $$(-1)^{j+k}\binom{i+j+k}{i,j,k}\prod_{\ell=0}^{k-1} \frac{i-j+k-2\ell-1}{i+j+k-2\ell-1}$$ apparently computes the ...
5
votes
2answers
114 views

Prove $((n+1)!)^n < 2!\cdot4!\cdots(2n)!$

so I know I need to prove this via induction, but I am somewhat stuck. Here is what I have does so far. Let $p(n) = (n+1)!^n \le 2!\cdot4!\cdot\ldots\cdot(2n)!$ $p(2) = 3!^2\le 2!\cdot4!$ Assume ...
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2answers
2k views

Summation of Series with Factorials

It is given that: $$v_n = n(n+1)(n+2)\;...\;(n+m)$$ $$and$$ $$u_n = (n+1)(n+2)\;...\;(n+m)$$ $i.$ Verify that: $$v_{n+1} - v_n = (m+1)(n+1)(n+2)\;...\;(n+m)$$ I started off by inspecting ...
0
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1answer
258 views

Finding modulus when all power of p are removed from N!

Given two integers $p$ and $N$. Let $m$ be number by $N!$ by max power of $p$ which divided $N!$. We have to find $m$ mod $p$. How to solve this?
12
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1answer
142 views

Solving $n!+m!+k^2=n!m!$ for positive integers $n,m,k$

I have been running in circles with this for a while now. It seems that the only solution is $(n,m,k)=(2,3,2)$ but I don't know how to prove it. Things I have noticed: WLOG $n\geq m$ we see that ...
19
votes
4answers
630 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
3
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0answers
40 views

Bound on Permutations [duplicate]

I am trying to prove the following inequality, $$n^{(l)} = n(n-1)\cdots(n-l+1) \geq \frac{n^l}{e}\quad\text{ for }\quad 2 \leq l \leq \sqrt{n}\;.$$ So my approach is to observe that $n^{(l)} = ...
3
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0answers
99 views

Prove (*) by induction on k.

Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form $$\sum_{i=1}^m ...
4
votes
3answers
269 views

Find all values of $n$ greater than or equal to 1 for which $n! + (n + 1)! + (n + 2)!$ is equal to a perfect square.

Not sure where to get started on this on. I started listing numbers for n starting at 1 but the numbers get very big very fast and I cannot find a pattern. Is there a better way of doing this or ...
7
votes
5answers
394 views

Find all solutions of the equation $x! + y! = z!$ [duplicate]

Not sure where to start with this one. Do we look at two cases where $x<y$ and where $x>y$ and then show that the smaller number will have the same values of the greater? What do you think?
4
votes
2answers
200 views

Is it significant that factorials have more trailing zeros as they get higher?

When I first learned about factorials in grade school I quickly became interested in the idea and did a lot of playing with them. I noticed, though, that as the factorials got higher and higher they ...
2
votes
4answers
4k views

Solving Equations with Factorials

I am attempting to solve an equation ${{n-2} \choose {2}} + {{n-3} \choose {2}} + {{n-4} \choose {2}} = 136$. With the formula for a combination being $\frac{n!}{r!(n - r)!}$, I simplified the given ...
5
votes
1answer
236 views

Generalization of the Factorial function

Is there any standard generalization of the Factorial function where the "skips" per multiplication is a parameter? For example, one generalization could be: $a(a-b)(a-2b)(a-3b)...1$ I tried to ...
5
votes
2answers
912 views

Ways to add up 10 numbers between 1 and 12 to get 70

I know this has something to do with factorials, and combinations and permutations. I've been puzzling over this for a little while, and I can't come up with an answer. My question is, How would one ...
2
votes
3answers
304 views

Prove that the combination formula can be reduced to…

Prove that: $$\frac{m!}{k!(m-k)!} = \frac{m}{k}\frac{m-1}{k-1}\cdots\frac{m-k+1}{1}$$ It's quite obvious when I write down some terms, but I just don't know how to make a rigorous proof. Any hints ...