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
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9answers
3k 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 ...
15
votes
4answers
3k 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 ...
13
votes
14answers
2k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
40
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?
4
votes
3answers
2k 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 ...
20
votes
6answers
4k 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 ...
1
vote
1answer
990 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 ...
6
votes
2answers
2k 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
2k 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 ...
6
votes
2answers
968 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 ...
8
votes
1answer
4k 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 ...
5
votes
4answers
212 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 ...
7
votes
5answers
574 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 ...
6
votes
2answers
220 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? $(p-0)!=0 \pmod{p}$ $(p-1)!=-1\pmod{p}$ $(p-2)!=1\pmod{p}$
5
votes
2answers
526 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?
4
votes
2answers
962 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} = ...
14
votes
6answers
3k 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 = ...
9
votes
2answers
2k 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 $100!$ ($5$ ...
12
votes
8answers
3k 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) = ...
9
votes
2answers
916 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 ...
8
votes
1answer
226 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
3answers
5k 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 ...
3
votes
4answers
12k 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.
25
votes
2answers
4k 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 ...
21
votes
9answers
3k 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 ...
9
votes
3answers
1k 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 ...
2
votes
0answers
60 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 $n > 1$, $x \ge 1$: ...
4
votes
1answer
4k 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 ...
2
votes
2answers
386 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 $n>2$.
1
vote
3answers
152 views

Integer ordered pairs $(x,y)$ for which $x^2-y!$…

[1] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2001$ [2] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2013$ My Try:: (1) $x^2-y! = 2001\Rightarrow x^2 = ...
32
votes
1answer
2k 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 ...
32
votes
14answers
3k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
11
votes
2answers
418 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 ...
22
votes
2answers
3k 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 ...
10
votes
3answers
1k views

Proof that $\frac{(2n)!}{2^n}$ is integer

I am trying to prove that $\dfrac{(2n)!}{2^n}$ is integer. So I have tried it by induction, I have took $n=1$, for which we would have $2/2=1$ is integer. So for $n=k$ it is true, so now comes time ...
8
votes
6answers
490 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 $n!$.
10
votes
3answers
138 views

$1!+2!+\ldots+n!$ cannot be the square of a positive integer

I have to prove that $1!+2!+\ldots+n!$ cannot be the square of a positive integer, $\forall n\geq4$. I've tried to do this with induction, but I don't seem to reach any satisfactory conclusion. Any ...
3
votes
3answers
480 views

Infinite Series using Falling Factorials

I recently started reading Concrete Mathematics by Graham, Knuth and Patashnik and met falling/rising factorials for the first time; it seemed like a very convenient method for evaluating particular ...
13
votes
6answers
748 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!), \ \ ...
5
votes
1answer
236 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) ...
1
vote
4answers
90 views

Find $\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}$ [duplicate]

I am having trouble showing $$\lim_{n\rightarrow\infty}\frac{n}{(n!)^{1/n}}=e.$$
2
votes
1answer
118 views

Is this induction procedure correct? ($2^n<n!$)

I am rather new to mathematical induction. Specially inequalities, as seen here How to use mathematical induction with inequalities?. Thanks to that question, I've been able to solve some of the form ...
10
votes
4answers
838 views

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
8
votes
4answers
846 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 ...
3
votes
2answers
64 views

Simple factorial function question

Show that $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+....$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove it . Any help will be greatly appreciated.
3
votes
1answer
1k 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?
1
vote
2answers
851 views

the nth root of 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 ...
1
vote
0answers
34 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
267 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
1answer
80 views

Falling factorial summands

Representing the summand as a falling factorial Compute the sum $$\sum_{k=1}^n\frac{1}{(k+1)(k+2)}$$