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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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 ...
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1answer
73 views

Evaluate the limit $\lim_{n\to \infty}{\frac{(n+3)!}{n^n}}$

Evaluate the limit $$\lim_{n\to \infty}{\frac{(n+3)!}{n^n}}, n\in \mathbb N$$ I know that the limit is $0$ but how to prove it?
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1answer
83 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: ...
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2answers
158 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 $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)(n+2)...(n+k)} = \frac{1}{kk!}$$
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4answers
109 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.
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0answers
47 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: ...
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1answer
186 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 $p$ ...
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4answers
159 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 $n\in\mathbb{N}$: $$(n+1)!=(n+1)\cdot n!$$ $$(0+1)!=(0+1)\cdot 0!$$ ...
6
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1answer
107 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 ...
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2answers
206 views

Compute largest integer power of $6$ that divides $73!$

I am looking to compute the largest integer power of $6$ that divides $73!$ If it was something smaller, like $6!$ or even $7!$, I could just use trial division on powers of $6$. However, $73!$ has ...
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2answers
76 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 ...
2
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2answers
442 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 $0.5!=\frac12\sqrt{\pi}$. ...
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0answers
125 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! + ...
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4answers
38 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 ...
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8answers
220 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, ...
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4answers
84 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.
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2answers
62 views

Equal products of consecutive integers

We have that $1\cdot2\cdot3\cdot4\cdot5\cdot6=8\cdot 9\cdot10$. An easy consequence is that $7!=7\cdot8\cdot9\cdot10$. I have been looking for more non trivial examples like these, but I have found ...
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1answer
42 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 ...
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2answers
249 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
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1answer
324 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 $n!$. ...
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3answers
115 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 ...
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3answers
38 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 ...
5
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2answers
122 views

Find the value of $\,\, \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 ...
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0answers
117 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 $$ ...
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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?
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2answers
126 views

Factorial Rational Limit

Anything besides the squeeze theorem. Here it is: $$\lim_{n\to\infty} \frac{(2n - 1)!}{{2n}^{n}}$$ Can someone start me off?
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1answer
78 views

Finding Factorial using Integral Definition

$n! = \int_{0}^{\infty} {e}^{-x}{x}^{n} \,dx$ How can we find $400!$? $400! = \int_{0}^{\infty} {e}^{-x}{x}^{400} \,dx$ Integration by parts is way too complicated, what are the other options?
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1answer
94 views

A limit with $((n-1)!)^{1/(n-1)}$ and other roots of factorials

How to prove that the following limit is positive? $$ \lim_{n \to \infty}\left(((n-1)!)^{1/(n-1)}-2\left(\frac{((n-1)!)^3}{(2n-2)!}\right)^{1/(n-1)}\right) >0,$$ where $ n\in \mathbb Z, n>1 ...
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0answers
38 views

Bounds on constant for Stirling approximation

Stirling's approximation says that $$n!\sim\sqrt{2\pi n}\left(\dfrac{n}{e}\right)^n.$$ What is known about constants $c_1$ and $c_2$ such that $$c_1\sqrt{n}\left(\dfrac{n}{e}\right)^n\le n!\le ...
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1answer
96 views

Show that $n!^{n+1}$ divides $(n^2)!$

My attempt so far is by induction. Let $f(n) = \frac{(n^2)!}{n!^{n+1}}$, I will try showing that $f(n)$ is a positive integer for all $n$. We have $f(0) = \frac{0!}{0!^{n+1}} = 1$. Now assume for ...
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0answers
94 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 ...
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1answer
36 views

Calculating the power of prime in factorial by changing base

The greatest power $k$ of a prime $p$ in the prime factorization of $n!$ is equal to $\frac1{p-1}(n-s(n)_p)$, where $s(n)_p$ is the sum of digits of $n$ when represented in base $p$. How to ...
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4answers
307 views

Closed Form for Factorial Sum

I came across this question in some extracurricular problem sets my professor gave me: what is the closed form notation for the following sum: $$S_n = 1\cdot1!+2\cdot2!+ ...+n \cdot n!$$ I tried ...
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1answer
54 views

Trouble understanding factorial algebra

I am having trouble understanding some of the algebraic concepts used here. In fact, the entire thing to me makes sense, except for the second red line. I don't understand how the diagonal swap ...
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1answer
60 views

Integration by parts with Legendre Functions

I need help deriving $\int_{-l}^l [P_l^m(x)]^2 = \frac{2}{2l+1} \frac{(l+m)!}{(l-m)!}$ for the associated Legendre functions I am supposed to use $P_l^m(x) = (-1)^{-m}\int_{-l}^l ...
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1answer
141 views

COmbinatoric : Guess who is the winner candidate?

National Radio Broadcast will put a contest to guess five winners out of twelve local boxers who will compete to win the best 5 boxers. All twelve boxers are equally good so the chance of winning is ...
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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?
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1answer
61 views

Summing an infinite series

I have been struggling with a problem involving a Markov Chain. To solve it I need to figure out the following ...
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1answer
93 views

Partial Derangements

There are n people and n houses, such that every person owns exactly one distinct house. Out of these n people, k people are special (k<=n). You have to send every person to exactly one house such ...
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2answers
50 views

Help with induction proof for recurrent function

I am having issues with the following inductive proof. Prove by induction on $n$ that $$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$ for all $n \geq 1,$ where ...
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0answers
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Query associated with Factorials and Series

I'm struggling to see how we go from Equation 1 to 2 to 3, which are seen below: Equation 1: $$P(Z=k) = p \binom{k/\delta-1}{n-1}(\lambda\delta)^{n}(1-\lambda\delta)^{k/\delta - n} $$ Equation 2: ...
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2answers
188 views

Summation of Binomial Theorem

The binomial theorem formula: $$\sum\limits_{k=0}^{n} {n \choose k} = \sum\limits_{k=0}^{n}\frac{n!}{k!(n-k)!} = \sum\limits_{k=0}^{n}\frac{n(n-1)(n-2) \cdots (n-k+1)}{k(k-1) \cdots 2\cdot1}.$$ I am ...
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2answers
57 views

Factorials and Mathematical induction

I'm having a bit of trouble understanding mathematical induction, particularly when there's a question with powers or factorials. For example I have a problem 1 x 1! +2 x 2! + 3 x 3! +... + n x n! = ...
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1answer
98 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 $$\lim_{n\to∞} \frac{f(n)}{\sum_{k=2}^n \frac1{\ln \ln(k!^{k!})}} = 1$$ ...
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1answer
47 views

n! mod c where c is a composite number

I am trying to write a program to calculate what is $n! \, \text{mod} \, c$, where $c$ is a composite number. While I understand $a b \, \text{mod} \, c$ is equal to $((a \, \text{mod} \, c) (b \, ...
18
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1answer
938 views

Can a double-factorial be a perfect square?

The title says it, basically. My question is $-$ for $ n \ge 2 $, can $n!!$ be a perfect square, where $!!$ represents the double-factorial? My conjecture is no, but I can't seem to be able to find a ...
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1answer
80 views

Finding the asymptotics of $\sum_{k=1}^n a^k k!$? Note that $a > 0$.

There's no way to use integration method in this case. I also tried to use Stolz–Cesàro theorem, but couldn't find right $y_n$. What method should I use?
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0answers
39 views

How are the Stirling-based bounds for the factorial function proven?

According to (26) on wolfram mathworld, one has $$\sqrt{2\hspace{-0.04 in}\cdot \hspace{-0.04 in}\pi} \cdot n^{n+(1/2)} \cdot \operatorname{exp}((-n)\hspace{-0.02 in}+\hspace{-0.02 ...
3
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2answers
57 views

Factorials Algebra

I have the following inequality $$5*10^{-10} \geq \dfrac{2^{n+1}}{(n+1)!}$$ Is there any way this can be solved algebraically? If not, is there a method that is better than guessing, for finding the ...
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1answer
248 views

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear?

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear? Attempt: Given 5 points, a line consist always of 2 points. Thus the total number of ...