Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.
10
votes
4answers
777 views
Highest power of a prime $p$ dividing $N!$
How does one find the highest power of a prime $p$ that divides $N!$ and other related products?
Related question: How many zeros are there at the end of $N!$?
This is being done to reduce ...
37
votes
3answers
1k views
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]
Possible Duplicate:
Highest power of a prime $p$ dividing $N!$
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
4
votes
9answers
1k views
Prove $0! = 1$ from first principles
How can I prove from first principles that $0!$ is equal to $1$?
3
votes
2answers
710 views
Derive a formula to find the number of trailing zeroes in $n!$ [duplicate]
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
I know that I have to find the number of factors of $5$'s, $25$'s, ...
1
vote
5answers
727 views
Why does 0! = 1? [duplicate]
Possible Duplicate:
Prove $0! = 1$ from first principles
Why does 0! = 1?
All I know of factorial is that x! is equal to the product of all the numbers that come before it. The product of ...
1
vote
1answer
424 views
Proof the inequality $n! \geq 2^n$ by induction
I'm having difficulity solving an exercise in my course.
The question is:
Prove that $n!\geq 2^n$.
We have to do this with induction. I started like this:
The lowest natural number where the ...
7
votes
5answers
483 views
Proving $\sum_{k=1}^n k\cdot k! = (n+1)!-1$ without using mathematical Induction. [duplicate]
Possible Duplicate:
Summation of a factorial
This equation is given:
$$
1\cdot1! + 2\cdot2! + 3\cdot3! + \ldots + n\cdot n! = (n+1)! - 1
$$
I've solved it using mathematical induction but ...
15
votes
6answers
2k views
prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer
Prove that $(2n)!/(n!)^2$ is even if $n$ is a positive integer. For clarity: the denominator is the only part being squared.
My thought process: The numerator is the product of the first n even ...
11
votes
8answers
995 views
How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$
It recently came to my mind, how to prove that the factorial grows faster than the exponential, or that the linear grows faster than the logarithmic, etc...
I thought about writing:
$$
a(n) = ...
5
votes
3answers
3k views
Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$
$$1 + \sum_{j=1}^{n} j!j$$
I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
4
votes
2answers
811 views
How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?
I have a couple things I want to prove. I'm pretty sure a proof by induction is the best route for these.
First, I need to show that $5^n < n!$ from some $n_{0} > 0$. I'm choosing $n_{0} = ...
20
votes
9answers
2k views
What is the purpose of Stirling's approximation to a factorial?
Stirling approximation to a factorial is
$$
n! \sim \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n.
$$
I wonder what benefit can be got from it?
From computational perspective (I admit I don't ...
7
votes
1answer
2k views
Last non Zero digit of a Factorial
I ran into a cool trick for last non zero digit of a factorial. This is actually a recurrent relation which states that:
If $D(N)$ denotes the last non zero digit of factorial, then
...
9
votes
2answers
405 views
Is there a way to solve for an unknown in a factorial?
I don't want to do this through trial and error, and the best way I have found so far was to start dividing from 1.
$n! = \text {a really big number}$
Ex. $n! = 9999999$
Is there a way to ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
11
votes
2answers
406 views
Given $n! = c$, how to find $n$?
I'm dealing with a time-complexity problem in which I know the running time of an algorithm:
$$t = 1000 \mathrm{ms} .$$
I also know that the algorithm is upper bounded by $O(n!)$.
I want to know ...
8
votes
6answers
444 views
Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$
Prove that for any $x \in \mathbb N$ such that $x<n!$ is the sum of at most $n$ distinct divisors of $n!$.
12
votes
2answers
1k views
Is there is any way to calculate generic formula for $1! +2! +3! + \cdots + n! $?
I came across a question where I need to find the sum of factorial of the first $n$ numbers.
So I was wondering if there is any generic formula for this?
Like there is a generic formula for different ...
12
votes
6answers
564 views
A question on the Stirling approximation, and $\log(n!)$
In the analysis of an algorithm this statement has come up:$$\sum_{k = 1}^n\log(k) \in \Theta(n\log(n))$$ and I am having trouble justifying it. I wrote $$\sum_{k = 1}^n\log(k) = \log(n!), \ \ ...
4
votes
1answer
133 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
4
votes
3answers
887 views
Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$
I need to check if
$$\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$$ converges or not. Additionally, I wanted to show that the sequence is monotonically increasing in n and so limit exists. Any help is ...
7
votes
2answers
150 views
Closed form for $(p-n)!\pmod{p}$ where $p$ is prime
Does $(p-n)!\pmod{p}$ have a closed form for any $n>2$ when $p$ is prime?
$(p-0)!=0 \pmod{p}$
$(p-1)!=-1\pmod{p}$
$(p-2)!=1\pmod{p}$
2
votes
2answers
326 views
How to prove $n < n!$ if $n > 2$ by induction?
I am stuck with the question below,
Prove by mathematical induction that $n<n!$ for $n>2$.
2
votes
1answer
341 views
What's the limit of the sequence $\lim_{n \rightarrow \infty} \frac{n!}{n^n}$?
$\displaystyle \lim_{n \rightarrow \infty} \frac{n!}{n^n}$
I have a question: Is it valid to use Stirling's Formula to prove convergence of the sequence?
1
vote
0answers
29 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 ...
1
vote
4answers
217 views
Finding $\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I'm looking for a way to find this limit:
$\lim_{n \to \infty} \frac{\sqrt{n!}}{2^n}$
I think I have found that it diverges, by plugging numbers into the formula and "sandwich" the result. However I ...
0
votes
0answers
42 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
0
votes
1answer
37 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 ...
29
votes
1answer
1k 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 ...
13
votes
6answers
2k 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$$ where $C$ is a constant. Maybe $C = ...
18
votes
4answers
359 views
Limit of series involving ratio of two factorials
$$
\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}
$$
The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
16
votes
5answers
691 views
How best to explain the $\sqrt{2\pi n}$ term in Stirling's?
I recently showed my Algorithms class how to bound $\ln n! = \sum \ln n$ by integrals, thereby obtaining the simple factorial approximation
$$ e \left(\frac{n}{e}\right)^{n} \leq n! \leq ...
18
votes
1answer
1k views
How many zeroes are in 100!
One common math puzzle I've seen around asks for how many zeros are in the product of "100!"
Usually, the solution everyone gives goes something like try to match pairs of 5s and 2s that factor out ...
17
votes
1answer
361 views
A series problem by Knuth
I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem.
Prove that $$\sum_{n=1}^\infty ...
15
votes
3answers
168 views
Proving that $\frac{(k!)!}{k!^{(k-1)!}}$ is an integer
I have to prove that:
$$\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}}$$
But I think I just made it harder, and ...
9
votes
1answer
107 views
Can this product be written so that symmetry is manifest?
Let $i,$ $j,$ $k$ be nonnegative integers such that $i+j+k$ is even. The expression
$$(-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}$$
apparently computes the ...
9
votes
5answers
645 views
How to prove that $\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2)… 2n} = \frac{4}{e}$
I'd like a hint to show that:
$$\lim \frac{1}{n} \sqrt[n]{(n+1)(n+2) \cdots 2n} = \frac{4}{e} .$$
Thanks.
7
votes
4answers
163 views
Combinatorial proof to $n! = (n-1)[(n-1)! + (n-2)!]$
It is for sure true that $n! = (n-1)[(n-1)! + (n-2)!]$
Since:
$(n-1)(n-1)! + (n-1)(n-2)! = $
$(n-1)(n-1)! + (n-1)! =$
$ (n-1)!(n-1+1) = (n-1)!n = n! $
Today my friend told me that there is a ...
7
votes
2answers
214 views
Finding all the numbers that fit $x! + y! = z!$
I have the formula $x! + y! = z!$ and I'm looking for positive integers that make it true. Upon inspection it seems that x = y = 1 and z = 2 is the only solution. The problem is how to show it.
...
4
votes
3answers
171 views
Factorial of infinity
So, I've read in this article that:
$$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$
And that:
$$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$
I found this result very ...
8
votes
2answers
256 views
Number of zeros not possible in $n!$ [duplicate]
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
The number of zeros which are not possible at the end of the $n!$ is:
...
7
votes
6answers
263 views
simplify summation of factorial (random walk)
I suspect that the expression
$$\sum_{n=0}^N \frac{(N-2n)^2}{n!(N-n)!}$$
simplifies to
$$\frac{2^N}{(N-1)!}$$
But I cannot find the intermediate steps. Can someone give me a hint how I can deduce ...
5
votes
2answers
183 views
Ways to add up 10 numbers between 1 and 12 to get 70
I know this has something to do with factorials, and combinations and permutations. I've been puzzling over this for a little while, and I can't come up with an answer. My question is, How would one ...
5
votes
3answers
164 views
Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)
I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate:
...
5
votes
2answers
369 views
Asymptotics of terms and errors in Stirling's Approximation
I have two related questions. Both are related to the asymptotics of Stirling's approximation, which is why I have included them in the same question. I will separate the questions if it is deemed ...
3
votes
1answer
55 views
Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?
In Ramanujan's proof of Bertrand's Postulate, Ramanujan states:
$\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$
where:
$\vartheta(x) = \sum_{p \le x} ...
2
votes
3answers
150 views
When is a factorial of a number equal to its triangular number?
Consider the set of all natural numbers $n$ for which the following proposition is true.
$$\sum_{k=1}^{n} k = \prod_{k=1}^{n} k$$
Here's an example:
$$\sum_{k=1}^{3}k = 1+2+3 = 6 = 1\cdot 2\cdot ...
8
votes
4answers
686 views
How many consecutive composite integers follow k!+1?
I wrote a program for myself in Mathematica to generate the answer for the first 300, which was really easy, but I can't find a pattern. The results are here. This is a problem in Underwood Dudley's ...
4
votes
1answer
74 views
What is the analytic continuation of a multifactorial?
The $\Gamma$ function is the analytic continuation of the factorial function. Is there a similar analog for multifactorials?
I am particularly interested in the double factorial. All Google has ...
