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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20
votes
5answers
5k 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 ...
45
votes
6answers
4k 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.
26
votes
12answers
9k views

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is ...
22
votes
15answers
4k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
27
votes
12answers
4k views

$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
13
votes
7answers
8k views

The product of n consecutive integers is divisible by n factorial

How can we prove that the product of n consecutive integers is divisible by n factorial? Note: In this subsequent question and the comments here the OP has clarified that he seeks a proof that ...
6
votes
3answers
7k views

The product of n consecutive integers is divisible by n! (without using the properties of binomial coefficients)

How can we prove, without using the properties of binomial coefficients, the product of n consecutive integers is divisible by n factorial?
7
votes
2answers
3k 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, ...
9
votes
5answers
481 views

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. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
4
votes
3answers
4k views

Limit of a sequence involving root of a factorial: $\lim_{n \to \infty} \frac{n}{ \sqrt [n]{n!}}$ [duplicate]

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 ...
27
votes
4answers
3k views

Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
21
votes
6answers
7k 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 ...
41
votes
3answers
2k 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?
5
votes
2answers
792 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?
19
votes
3answers
2k views

$n!+1$ being a perfect square

One observes that \begin{equation*} 4!+1 =25=5^{2},~5!+1=121=11^{2} \end{equation*} is a perfect square. Similarly for $n=7$ also we see that $n!+1$ is a perfect square. So one can ask the truth of ...
3
votes
5answers
4k 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 ...
3
votes
2answers
2k 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 ...
9
votes
4answers
28k views

Derivative of a factorial

What is ${\partial\over \partial x_i}(x_i !)$ where $x_i$ is a discrete variable? Do e consider $(x_i!)=(x_i)(x_i-1)...1$ and do product rule on each term, or something else? THanks.
3
votes
7answers
202 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 ...
18
votes
7answers
4k 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 = ...
13
votes
4answers
4k 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 ...
6
votes
2answers
2k views

On the factorial equations $A! B! =C!$ and $A!B!C! = D!$

I was playing around with hypergeometric probabilities when I wound myself calculating the binomial coefficient $\binom{10}{3}$. I used the definition, and calculating in my head, I simplified to this ...
9
votes
1answer
5k 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 ...
3
votes
5answers
6k views

Calculating the limit $\lim((n!)^{1/n})$

Find $\lim((n!)^{1/n})$. The question seemed rather simple at first, and then I realized I was not sure how to properly deal with this at all. My attempt: take the logarithm, $$\lim \ln((n!)^{1/n}) = ...
6
votes
2answers
259 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? $(p-0)!=0 \pmod{p}$ $(p-1)!=-1\pmod{p}$ $(p-2)!=1\pmod{p}$
27
votes
2answers
8k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
13
votes
8answers
6k views

How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$ [duplicate]

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) = ...
7
votes
3answers
7k views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$ [duplicate]

$$1 + \sum_{j=1}^{n} j!j$$ 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 ...
5
votes
1answer
3k views

Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?

I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq ...
8
votes
5answers
627 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 ...
1
vote
2answers
84 views

Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in ...
1
vote
3answers
2k views

limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
56
votes
1answer
3k 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 ...
9
votes
2answers
4k views

Number of zero digits in factorials

Here is a riddle someone has been asked in a job interview: How many zero digits are there in $100!$? Well, I found the first $24$ quite fast by counting how many times five divides $100!$ ($5$ ...
6
votes
1answer
10k views

Is there a way to reverse factorials? [duplicate]

Possible Duplicate: Is there an Inverse Gamma $\Gamma^{-1} (z) $ function? Is there any way I can 'undo' the factorial operation? JUst like you can do squares and square roots, can you do ...
5
votes
2answers
1k 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} = ...
4
votes
3answers
146 views

Evaluating $\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$ [duplicate]

$$\lim_{n\to \infty}\frac1{2n}\log\left({2n \choose n}\right)$$ Now, $\log(n!) = \Theta (n\log(n))$ so I think we could write, $$\lim_{n\to\infty}\frac{1}{2n}\left(\log\left(2n!\right) - ...
29
votes
1answer
6k 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 ...
11
votes
2answers
2k 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 ...
11
votes
2answers
347 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. ...
9
votes
3answers
2k views

How to define fractional factorials, like 3.6!? [duplicate]

I did not know that you could find an answer for that. However, I can only use Excel so far to do it. How to calculate 3.6! by hand?
4
votes
1answer
237 views

To find all $n$ such that $(n-1)!+1$ is a perfect power of $n$

How to find all positive integers $n$ such that $(n-1)!+1$ can be written as $n^k , k\in \mathbb Z^+$ ?
8
votes
1answer
253 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 ...
5
votes
4answers
418 views

Evaluate $\sum_{k=1}^nk\cdot k!$

I discovered that the summation $\displaystyle\sum_{k=1}^n k\cdot k!$ equals $(n+1)!-1$. But I want a proof. Could anyone give me one please? Don't worry if it uses very advanced math, I can just ...
4
votes
3answers
125 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
7
votes
1answer
3k 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?
2
votes
6answers
114 views

Prove by induction that $n^2<n!$

How can I show that $n^2<n!$ for all $n\geq 4$ Step 1 For $n=1$, the LHS=$4^2=16$ and RHS=$4!=24$. So LHS$<$ RHS. Step 2 Suppose the result be true for $n=k$ i.e., $k^2<k!$ Step 3 ...
1
vote
2answers
188 views

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

How does one prove by induction that $n! > n^2$ for $n \geq 4$
0
votes
1answer
68 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?
42
votes
11answers
25k views

Do factorials really grow faster than exponential functions?

Having trouble understanding this. Is there anyway to prove it?