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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2
votes
2answers
55 views

year 10 factorial question

I would like to know the number of zeros occuring in the factorial of 2016? (2016!) I have read some ways but i don't understand it.
35
votes
17answers
8k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
0
votes
0answers
62 views

Accurately calculating large factorials

I need to calculate the exact value of $1,000,000,000!$ It can't be an approximation. As far as I am aware, the only way to do this would be to calculate it starting from the beginning $(1\cdot2\cdot3\...
6
votes
3answers
232 views

Seeking closed form for infinite sum $\sum \limits_{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$

$\displaystyle \sum _{ n=1 }^{ \infty }{ \frac { { \left(n! \right) }^{ 2 } }{ { n }^{ 3 }(2n)! } }$ is approximately $.5229461921333351$ but I've been assured that there is a closed form for this ...
11
votes
2answers
193 views

Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$

Does a closed form exist for $$\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$$ in terms of $k$ and other functions? The best that I have been able to do is solve the case where $k=1$, since the ...
6
votes
3answers
155 views

Find the value of : $\lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} $

First of all, sorry if something similar to this has been posted before (it's my first time in this web). I need to calculate the limit as $n\rightarrow \infty$ for this: $$\lim\limits_{n\to \infty} \...
2
votes
1answer
41 views

Do negative binomials imply negative factorials exist?

I've seen the following identity: $$\binom{-n}{k} = (-1)^k\binom{n+k-1}{k}$$ So I tried to derive it, assuming negative factorial was a real concept, having it extend down to negative infinity: $$\...
2
votes
2answers
321 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
2answers
60 views

simplify factorials: $\frac{(k-1)!}{(k+2)!}$ [duplicate]

Question: simplify $$\frac{(k-1)!}{(k+2)!}$$ What I did was: $$\frac{(k - 1)!k!}{(k + 2)! \cdot (k + 1)!}$$ This I did following the rule $n! = n \times (n - 1)!$. can this be simplified ...
0
votes
1answer
15 views

Prove that: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ [duplicate]

Question asks to prove: ${^{n}\mathrm{C}_{k}} = {^{n-1}\mathrm{C}_{k-1}}+{^{n-1}\mathrm{C}_{k}}$ My Steps: $$\begin{align*}\frac{(n-1)!}{(n-k-2)!(k-1)!} + \frac{(n-1)!}{(n-k-1)!(k)!} & = \...
3
votes
2answers
80 views

Finite summation including binomial coefficients and double factorials

I came across the following summation: $$ \sum_{k=0}^n\frac{(-1)^k(2k)!!}{(2k+1)!!}\dbinom{n}{k}\,\,\,\,(n\in\mathbb{N}). $$ $\tbinom{n}{k}$ are binomial coefficients, $n!/k!(n-k)!$. Mathematica told ...
3
votes
1answer
51 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. ...
1
vote
1answer
33 views

Simplifying $\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$

Question asks to simplify: $$\frac{^n\mathrm{C}_k}{^n\mathrm{C}_{k-1}}$$ I have a few steps but not sure if its correct. $$\begin{align*}\frac{(n)!}{(n-k)!(k)!} \bigg/ \frac{(n)!}{(n-k-1)!(k-1)!}...
0
votes
2answers
64 views

Simplifying factorials: $\frac{(n-1)!}{(n-2)!}$

Question: simplify $$\frac{(n-1)!}{(n-2)!}$$ What I did was: $$\frac{(n - 1)!}{(n - 2)! \times (n - 3)!}$$ This I did following the rule $n! = n \times (n - 1)!$. But my answer just doesn't look ...
4
votes
1answer
25 views

Find an explicit map with certain combinatorial properties

The following combinatorial problem popped up in a totally uncombinatorial context: Let $\mathcal{P}$ denote the power set of a set and let $k \in \mathbb{N}$. Is there a map $c: \mathcal{P}(\{1,2,\...
1
vote
2answers
28 views

Analytic continuation and how it relates to the gamma function.

I am familiar with factorials, and I have read about the gamma function. From what I understand, the gamma function extends the concept of the factorial to complex numbers by nature of being an ...
1
vote
1answer
17 views

Find a map on a power set with certain combinatorial properties

The following combinatorial problem popped up in a totally uncombinatorial context: Let $\mathcal{P}$ denote the power set of a set and let $k \in \mathbb{N}$. Is there a map $c: \mathcal{P}(\{1,2,\...
5
votes
2answers
187 views
+50

Find last 5 significant digits of 2017!

Since there are less powers of $5$ than of $2$ and since $10 = 2 \cdot 5$, I counted the number of zeros in $2017!$: $\left \lfloor{ \frac{2017}{5^1}}\right \rfloor + \left \lfloor{ \frac{2017}{5^2}}\...
1
vote
0answers
97 views

Counting zeros in a factorial(terminal + zeros in between digits)

The usual questions involving counting zeros in a factorial asks us to count only the terminal zeros. This question asks to count the zeros that are in between digits, for example, 8! (40320, has a ...
4
votes
1answer
57 views

Irrationality of the concatenation of the rightmost nonzero digits in $n!$

Surfing the internet I bumped into a very interesting problem, which I tried to solve, but got no results. The problem is following: let $h_n$ be the most right non-zero digit of $n!$, for example, $...
0
votes
1answer
19 views

recurrence relation - How to determine pattern for an even or odd or different type of factorial

Hi I am having trouble on how to solve for the odd terms of recurrence relation in terms of exponential and factorials. How are you able to see a pattern to simplify a non standard factorial. This ...
2
votes
1answer
56 views

Is there a more concise expression of this product?

In a longer computation, I have stumbled upon the following product, where $k,r \in \mathbb{N}_0$ are fixed numbers: $$\prod_{0 < i_0<i_1<\dots<i_r\leq k} (i_r-i_{r-1})(i_{r-1}-i_{r-2})\...
8
votes
1answer
442 views

Conjecture about primes and the factorial: for all primes $p>5$, must there exist a prime $q<p$ such that $q\equiv m!\pmod p$ for some $2<m<p$?

Below $0\notin\mathbb N$. Further corrected conjecture: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, $2<m<p$. or Given a prime ...
11
votes
6answers
779 views

Expressing a factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. Here's the formula: $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$ Can anyone give a proof of this result? Note:...
19
votes
5answers
922 views

A strange combinatorial identity: $\sum\limits_{j=1}^k(-1)^{k-j}j^k\binom{k}{j}=k!$ [duplicate]

In reading about A polarization identity for multilinear maps by Erik G F Thomas, I am led to prove the following combinatorial identity, which I cannot find anywhere, nor do I have any idea how to ...
2
votes
3answers
53 views

Proof by induction: inequality $n! > n^3$ for $n > 5$

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
2
votes
1answer
40 views

notation for factoraling a factorial? (since one cannot do n!!)

I was thinking about how to get a number to be larger than graham's number very easily... came up with "factoraling" a factorial. However the notation n!! means something completely different. And I ...
6
votes
4answers
166 views

Proof of the summation $n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$?

I was going through a Number Theory book the other day and found this question. It asked for the proof of the following equation: $$n!=\sum_{k=0}^n \binom{n}{k}(n-k+1)^n(-1)^k$$ I tried hard but ...
2
votes
2answers
74 views

Number of zeros at the end of $10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$

How do I find the number of zeros at the end of the Integer $$10^{2}!+ 11^{2}!+12^{2}! \cdots+99^{2}!$$ Answer provided for this question is $24$
1
vote
0answers
23 views

When is $\frac{2 n f(n)}{n !}$ in the order of some fixed power of $n$?

I would like to know when $\frac{2 n f(n)}{n !}$ is $O (n^b)$ where $b$ is a constant. Here, $n$ is a positive integer. My attempt: $$ \frac{2 n f(n)}{n !} = \frac{2 n f(n)}{\sqrt{2 \pi n} (\frac{n}{...
9
votes
4answers
138 views

Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
0
votes
2answers
78 views

Factorial sum estime

Prove that: $$\displaystyle \sum_{n=m+1}^\infty \dfrac{1}{n!} \le \dfrac{1}{m\cdot m!}$$ I have tried induction on $m$ but it does not work very well. Any suggestion?
5
votes
3answers
90 views

Factorial Proof by Induction Question? [duplicate]

$\text{Use the PMI to prove the following for all natural numbers n.}$ $ \frac{1}{2!} + \frac{2}{3!} + \cdot \cdot \cdot + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!} $ So for this question I get ...
2
votes
1answer
130 views

A couple of series questions that I just can't figure out (Calc 2)

Show that $$ \begin{align} \left(\frac{\pi}{2}\right)^2\left[\int_0^{\pi/2}\cos^{2n}t\ dt-\int_0^{\pi/2}\cos^{2n+2}t\ dt\right]&=\frac{\pi^3}{8}\left[\frac{(2n-1)!!}{(2n)!!}-\frac{(2n+1)!!}{(2n+...
5
votes
5answers
1k views

Approximation of log(n!)

I just finished calculus 1 (derivative and integral) then I take another course on calculus 2. In the video the professor talks about the the series $$\frac{n!}{(\frac{n}{e})^n}$$ He shows the ...
1
vote
1answer
54 views

Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. [duplicate]

This question is inspired by Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$. Find all nonnegative integers $m$ and $n$ such that $m!+1=n^2$. We know that $(m,n)=(4,5)$, $(...
6
votes
1answer
245 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how can ...
11
votes
4answers
2k views
1
vote
1answer
25 views

Finding the remainder of an exhausted $m!$

For any prime $p$, let $n_p(m)$ denote the exponent of $p$ in the factorisation of $m!$, i.e. $m!=p^{n_p(m)}\cdot k$ with $p\not\mid k$. Is wonder if there is a general formula of $\frac{m!}{p^{n_p(...
1
vote
1answer
31 views

From $ \frac{\left(n\cdot \:n!+1\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)!+1} $ to $ \frac{n+\frac{1}{n!}}{n+1+\frac{1}{n!}} $?

Good evening to everyone. I have an expression that I don't know how to arrive at. $$ \frac{\left(n\cdot \:n!+1\right)\left(n+1\right)}{\left(n+1\right)\left(n+1\right)!+1} $$ to $$ \frac{n+\frac{1}{n!...
0
votes
0answers
78 views

Closed-from for the series: $\sum_{k=0}^{\infty} \frac{1}{(k!)!}$

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don't mean the double factorial (treated here) ...
2
votes
1answer
143 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 ...
1
vote
1answer
62 views

Is there a closed-form for $\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$?

As the title says, I'm wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)^2}$$ Trying on WolframAlpha, I get the value $2....
77
votes
10answers
8k views

Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
0
votes
1answer
24 views

How to simplify modular exponential expressions with factorial as exponents?

Said I have the following expression: n = 39^50! mod 2251 By Fermat's Little Theorem: 39^2250 = 1 mod 2251 Solving: 2250 = 50.45 n = 39^50.49.48.47.46.45.44! mod 2251 Let b = 49.48.47.46.44! , ...
5
votes
2answers
168 views

Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
1
vote
1answer
101 views

How to prove $\frac{(2n)!(2m)!}{n!m!(n+m)!}$ is an integer by strictly using my method?

I have to prove that $$\frac{(2n)!(2m)!}{n!m!(n+m)!}$$ is always an integer. I already have seen the same question here-Prove that for all non-negative integers $m,n$, $\frac{(2m)!(2n)!}{m!n!(m + n)!}...
0
votes
2answers
91 views

Prove that $\log(n!)\leq(\log(n))!$

Prove that $\log(n!)\leq(\log(n))!$ My attempt: I read somewhere that $n\leq\log(n!)\leq(\log(n))!$. But when I used calculator $\log(n!)$ can not be less than or equal to $(\log(n))!$. ...
1
vote
0answers
24 views

Function order of Logarithms

How can I use Stirling's approximation to trying to find the function order of $ceil(log(logx))!$ ? My main goal is to finding it's order of complexity but my main issue is that I'm not sure on how to ...
1
vote
5answers
90 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...