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
votes
2answers
145 views

Digit in units place of $1!+2!+\cdots+99!$

There isn't much I can add to the question description to expand upon the title. I came across this in a multiple choice test. The options were $3$, $0$, $1$ and $7$. I am absolutely stumped. Any ...
0
votes
1answer
22 views

Limit of factorials

I'm failing go figure out how to calculate the limit where I have one factorial divided by two at about half its size. The specific limit I'm trying to find is this: $$\lim_{n\to \infty}\frac ...
2
votes
0answers
79 views

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$

Find $\displaystyle\sum_{k=1}^\infty\dfrac{1}{\left(\binom{2014+k}{2014}\right)}$, where $\left(\binom{a}{b}\right)=\dfrac{a!!}{b!!(a-b)!!}$ EDIT : Someone pointed out in the Mathematics chat that my ...
1
vote
2answers
36 views

for all positive integers m there exists consecutive primes which are at least m apart

I'm having difficulty as to how I should approach this problem, any help would me much appreciated! Note that $k$ divides $n! + k$ for each $k\le n$. Use this fact to show that for all positive ...
1
vote
1answer
17 views

Prove that if $p\le n$, then $p$ does not divide $n! + 1$

I'm having trouble on how to approach this problem Prove that if $p\le n$, then $p$ does not divide $n! + 1$ ($p$ is prime and $n$ is an integer).
0
votes
0answers
34 views

Inequality involving factorial and a number 1/12

How I can prove the following two inequalities: If $n$ is a positive integer then $$ \sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n+1}}<n!<\sqrt{2 \pi}n^{n+\frac{1}{2}}e^{-n+\frac{1}{12n}} $$
22
votes
11answers
6k 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 ...
2
votes
3answers
145 views

What is the practical application of factorials

I'm trying to understand the practical application of factorial - in simple applications. I searched the math.stackexchange and could not find an answer. I understand that a factorial of n items ...
0
votes
4answers
84 views

Why is ${{n+1}\choose{k}}={{n}\choose{k-1}}+{{n}\choose{k}}$? [duplicate]

My teacher showed us a proof by induction for this equation for $n\in\mathbb{N}$: $$\sum\limits_{k=0}^n{{n}\choose{k}} = 2^n$$ In the first step, this sum is rewritten using ...
-1
votes
4answers
54 views

How can we find factorials in decimal form? [duplicate]

I've heard of factorials such as $5!$ and $3!$, which work like this: $5!=5\times4\times3\times2\times1=120$ and $3!=3\times2\times1=6$. At least this is what we get. Also, surprisingly, $0!=1$, but ...
0
votes
0answers
34 views

How come negative factorials never give us an answer?

I've done this and it always gave me an error probably because of this (it'll continue):$$4!=4*3*2*1=24$$$$3!={4!\over 4}={24\over 4}=6$$$$2!={3!\over 3}={6\over 3}=2$$$$1!={2!\over 2}={2\over ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
2
votes
4answers
62 views

A question about sums and factorials

Consider the sum $S=x!+\sum_{i=0}^{2013}i!$, where $x$ is a one-digit nonnegative integer. How many possible values of $x$ are there so that S is divisible by 4?
5
votes
4answers
286 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 ...
8
votes
10answers
165 views

How to evaluate $\lim_{n\to\infty}\sqrt[n]{\frac{1\cdot3\cdot5\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot6\cdot\ldots\cdot2n}}$

Im tempted to say that the limit of this sequence is 1 because infinite root of infinite number is close to 1 but maybe Im mising here something? What will be inside the root? This is the sequence: ...
2
votes
3answers
79 views

To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 … (2n-1)\over 2.4.6…(2n)$

How can I show $\lim_{n \to \infty} a_n = 0$ $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$ I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below ...
4
votes
3answers
48 views

Calculating the nth derivative of $\frac{x}{x+1}$

I was asked to calculate the nth derivative of $f(x) =\frac{x}{x+1}$. My solution: $$ f'(x) = (x+1)^{-2}$$ $$f''(x) = (-2)(x+1)^{-3}$$ $$f'''(x) = (-2)(-3)(x+1)^{-4}$$ $$f^{n}(x) = n!(x+1)^{-(n+1)} . ...
2
votes
2answers
180 views

Limit of a fraction of double factorials

How can we show that $\begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*}$ where $\begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*}$ ...
2
votes
3answers
33 views

factorial division when the bottom number is larger than the top number

I have a factorials problem to solve, and I do not know the method of solving it. I know how to do one number factorials (e.g. 5!, 15! etc...) and factorial division where the top number is larger ...
1
vote
2answers
326 views

Is there a rule to the terms of a falling factorial?

$\require{cancel}$I discovered that $n!=\xcancel{(n)_{n-1}}n^{\underline{n-1}}=n(n-1)(n-2)\cdots(3)(2)$. I have expanded a few examples: $$2!=\xcancel{(2)_1}2^{\underline{1}}=2\\ ...
7
votes
5answers
388 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?
1
vote
1answer
22 views

Factorial Divides Rising Power Proof Help

I'm trying to prove the following: $m^{\overline n} \equiv 0 \bmod n!$ Where $m^{\overline n} = m\left({m+1}\right)\left({m+2}\right)\ldots\left({m+n-1}\right)$, the product of $n$ successive ...
3
votes
1answer
100 views

How to prove that $\sum\limits_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)…(n+k)} = \frac{1}{kk!}$ for every $k\geqslant1$

Does anyone have any idea how to prove that $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$$
4
votes
2answers
81 views

Prove that $\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+a)}=\frac{1}{aa!}$ [duplicate]

$$\sum_{n=1}^{\infty}\frac{1}{n(n+1)\cdots(n+v)}=\frac{1}{vv!}$$ I am struggling to find a solution for this but no luck yet. How can I analyze it to get to second part?
0
votes
0answers
51 views

Relation between Hyperfactorial, Superfactorial, Pascal's Triangle and Binomial Coefficient

I read here that the product of the elements in the $N^{th}$ row of Pascal's triangle is equal to $(n!)^{n+1}/(\prod_{k=1}^n k!)^2$. Let's call the product of elements in the $i^{th}$ row of Pascal's ...
0
votes
1answer
23 views

Rising factorial power

How the expression below can ve proved: $(a + b)^{\overline{n}} = \sum\limits_{j=0}^{n}C_n^j a^{\overline{n-j}}b^{\overline{j}}$ where $x^{\overline{n}}$ - is rising factorial power: ...
1
vote
4answers
84 views

Show that $(k!)^n$ divides $(kn)!$

Show that $(k!)^n$ divides $(kn)!$ I've tried it but without success. Any help would be great.
1
vote
0answers
34 views

Calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$

We know that $\lim_{n\rightarrow \infty}\frac{n_1}{n}=p$ and $0\leq p\leq 1$. Based on this information I want to calculate $\lim_{n\rightarrow \infty}\frac{\log{\binom{n}{n_1}}}{n}$. Any help? Note: ...
11
votes
1answer
153 views

Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2

Prove that: If $n$ is greater than 2, then $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ From Barnard & Child's "Higher Algebra". I know that the highest power of a prime $p$ ...
1
vote
3answers
79 views

Could negative integer factorials be defined in some way?

I know that, calling $F$ such an extension, if we wanted to keep having $$ F(z+1)=(z+1)F(z),$$ letting $ z=-1$ would lead to the absurdity $ 1=0$. Also, $ \Gamma(z)$ has poles at $ z \in ...
1
vote
4answers
153 views

Is this a demonstration or a definition?

Some people says that this can be demonstrated $0!=1$, but other say that this is a definition. Which one is correct? Let's given $n\in\mathbb{N}$: $$(n+1)!=(n+1)\cdot n!$$ $$(0+1)!=(0+1)\cdot 0!$$ ...
3
votes
0answers
65 views

Proving that $\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = {1\over\pi}$

While trying to prove that $$(1)\qquad x\sum_{k=0}^\infty\frac{2^{-5k}(6k+1)((2k-1)!!)^3}{4(k!)^3} = 1 \implies x=\pi$$ I got to a point, using W|A, where I have to prove that $$\color{red}{(2)\qquad ...
4
votes
2answers
47 views

Is there a Sum Factorial?

I am curious if there is any addition factorial. Obviously, $$x! = \prod_{n=1}^x n$$ but what I want is a shorthand way of writing: $$\sum_{n=1}^x n$$ So is there such a thing? and if so, what is ...
1
vote
2answers
70 views

How to calculate decimal factorials, like $0.78!$ [duplicate]

When I enter $0.78!$ in Google, it gives me $0.926227306$. I do understand that $n! = 1\cdot2\cdot3 \cdots(n-1)\cdot n$, but not when $n$ is a decimal. I also have seen that $0.5!=\frac12\sqrt{\pi}$. ...
5
votes
0answers
81 views

Primality of $n! +1$

I came across with a problem where I was required to examine primality of $n! +1$ (17! + 1 was the actual number). Although Wilson's Theorem could be manipulated for determining primality of $n! + ...
11
votes
2answers
643 views

What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?

What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$. I have tried to find the answer using the Binomial Theorem but that doesn't help. How will we do this? Please help.
0
votes
4answers
30 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
7
votes
1answer
3k views

When does a Square Matrix have an LU Decomposition?

When can we split a square matrix (rows = columns) into it’s LU decomposition? The LUP (LU Decomposition with pivoting) always exists; however, a true LU decomposition does not always exist. How do ...
3
votes
1answer
85 views

How to find asymptotics of this sum

Is there any way to find $f(n)$ in this term: $$\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})} \sim f(n)?$$ The tilde symbol means that $$\lim_{n\to∞} \frac{f(n)}{\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})}} = 1$$ ...
0
votes
8answers
209 views

Prove that $n! > n^5$

I'm trying to prove that $n! > n^5$ for large enough values of n. While it seems obvious that this should be true, I have no idea how to prove it rigorously. EDIT: So, looking at the comments, ...
3
votes
4answers
82 views

How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
0
votes
1answer
28 views

Factorials with fractions

I don't understand how $$ \frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!} $$ becomes $$ \frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)} $$ and then how it becomes $$ \frac {n+n^2+n}{n-1} $$ I've tried applying ...
0
votes
2answers
74 views

How to evaluate factorials greater than $69!$

How to evaluate factorials greater than $69!$? On my calculator, $69!$ is the largest number I can enter before it gives me a syntax error, most likely due to an overflow. Is there a way to evaluate ...
4
votes
1answer
112 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
4
votes
2answers
97 views

Find the limit $\,\,\, \lim_{n \to \infty}\Big(\big(1+\frac{1}{n}\big)\big(1+\frac{2}{n}\big) \cdots\big(1+\frac{n}{n}\big)\Big)^{1/n} $

What is the limit of: $$ \lim_{n \to \infty}\bigg(\Big(1+\frac{1}{n}\Big)\Big(1+\frac{2}{n}\Big) \cdots\Big(1+\frac{n}{n}\Big)\bigg)^{1/n}? $$ By computer, I guess the limit is equal to ...
0
votes
1answer
45 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?
0
votes
0answers
43 views

Nearest factorial given a number.

Hi suppose I have given a number lets say 344545.Is there a way to determine he nearest smallest factorial? This is a Multiple choice question(one question 4 option).So what can be the fastest ...
0
votes
3answers
36 views

$\frac{(n/2)!}{n!} = \frac{1}{2^{n/2}(n-1)!!}$?

I was working on a puzzle involving some rather complex probability when I arrived at two very distinct methods with very different ways of calculating the probability of solving the puzzle. The ...
2
votes
2answers
63 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
1
vote
0answers
48 views

Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ ...