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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11
votes
4answers
16k views

Factorial, but with addition [duplicate]

Is there a notation for addition form of factorial? $$5! = 5\times4\times3\times2\times1$$ That's pretty obvious. But I'm wondering what I'd need to use to describe $$5+4+3+2+1$$ like the ...
2
votes
1answer
115 views

Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

I have to check for which $x$ the series converges/diverges. $\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$ I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > ...
1
vote
1answer
276 views

In how many ways can five balls be chosen so that…

In how many ways can five balls be chosen so that (a) two are red and three are black? (b) three are red and two are black? out of $7$ black and $8$ red Should I use permutation? or ...
2
votes
3answers
117 views

Limit of sequence. with Factorial

Can't find the limit of this sequence : $$\frac{3^n(2n)!}{n!(2n)^n}$$ tried to solve this using the ratio test buy failed... need little help
9
votes
5answers
457 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
2answers
110 views

Show that $p!$ and $(p - 1)! - 1$ are relatively prime

If $p$ is prime number, with $p>3$ Show that $p!$ and $(p - 1)! - 1$ are relatively prime. I tried $\text{gcd}\;(p!,(p-1)!-1)=d\Longrightarrow d\mid p!$ e $d\mid(p-1)!-1$ having ...
0
votes
3answers
180 views

how to solve factorial involving multiplication

I am trying to solve this question but not able to find any helpful material. It involves factorial with multiplications, $$\frac{8!}{5!}\cdot \frac{7!}{7!10!}$$ I tried crossing 8 and 5 and 7 with ...
0
votes
1answer
49 views

Review of an answer for finding a limit of a sequence

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {{n!} \over {(n + 1)(n + 2)...(2n)}} = {{n!} \over {{{(2n)!} \over {n!}}}} = \cr & {{n!n!} \over {2n!}} = {{n!} \over 2} = + \infty ...
17
votes
3answers
450 views

Making $121$ with five $0$s

So I say this puzzle online a few days ago and found it quite interesting. The original question was Make $120$ using only five $0$s. Well, I said to myself, this is utterly trivial. Note that ...
2
votes
2answers
266 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
2answers
327 views

Convergence testing involving factorial and square root

I'm trying to find the convergence of this using the ratio test: $$\displaystyle\sum_{i=1}^{\infty}\dfrac{1}{\sqrt{t!}}.$$ But I'm getting no luck! Can anyone help? (sorry I've not quite ...
3
votes
2answers
105 views

Prove that $\sum_{k=0}^n\frac{1}{k!}\geq \left(1+\frac{1}{n}\right)^n$ [duplicate]

It basically says it all in the title. I tried solving the inequality using the bernoulli inequality somehow $$\dfrac{\displaystyle\sum_{k=0}^n\frac{1}{k!}}{(1+\frac{1}{n})^n}\geq 1,$$ but the ...
1
vote
1answer
33 views

Is there an actual expansion of the Gamma function's integral?

$$\int_0^{\infty} x^{t-1} e^{-x} \, \mathrm{d}x = (t-1)! = \Gamma (t)$$ Is the expression $(t-1)!$ the actual result of integrating the gamma integral? Meaning, if you were to compute the integral ...
3
votes
1answer
150 views

Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

Stirling's approximation of the factorial for even numbers is given by $$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$ Further, the Euler numbers grow quite rapidly for large ...
1
vote
1answer
80 views

Limit of a function not using Stirling's Approximation

I want to compute the following limit: $$\lim_{n\to\infty} \frac{\left(\frac{e}{F_{n+1}}\right)^{F_{n+1}} F_{n+1}!}{\left(\frac{e}{F_n}\right)^{F_n} F_n!},$$ where $F_n$ is the $n$th Fibonacci ...
1
vote
3answers
152 views

Solve Algebraical.ly $0.5=\dfrac{365!}{365^{n}(365-n)!} $

How does one go about solving this equation? Not sure how to approach this as no factorials will cancel out. Im sorry I meant $\dfrac{365!}{365^{n}(365-n)!}=0.5$.
0
votes
3answers
59 views

Is $\sum_{n=1}^{\infty}\frac{2^nn!}{(n+1)!}$ absolutely convergent?

I'm very uncomfortable with factorials just because I haven't done many of them. But my basic understanding is if I start with (for example) $(n+1)!$ then this is equivalent to $(n+1)*(n)$ and if it ...
6
votes
3answers
149 views

Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$

I am trying to calculate the following limit without Stirling's relation. \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you ...
2
votes
1answer
101 views

Proving a sequence with induction reasoning

I have an assignment which I am quite stuck on. The question is the following: function f: N to N is defined recursivly: ...
0
votes
1answer
37 views

Series — Coefficient Cn and Radius of Convergence

. I'm lost, and my textbook is failing me
3
votes
3answers
275 views

Use a factorial argument to show that $C(2n,n+1)+C(2n,n)=\frac{1}{2}C(2n+2,n+1)$

I need some help, showing that the left hand side is equivalent to the right hand side. I tried but I get stuck, I am not sure if I am on the right path. Here is my attempt: $C(2n,n+1) + C(2n,n)$ ...
3
votes
5answers
244 views

Need help understanding the factorial formula $n!=n(n-1)(n-2)\cdots(3)(2)(1)$

The caption says the following: If $n$ is an integer such that $n \ge 0$ then $n$ factorial is defined as, $$n!=n(n-1)(n-2)\cdots(3)(2)(1)$$ if $n \ge 1$ by definition. I'm really just confused by ...
3
votes
2answers
793 views

Factorial lower bound

A professor in class gave the following lower bound for the factorial $$ n! \ge {\left(\frac n2\right)}^{\frac n2} $$ but I don't know how he came up with this formula. The upper bound of $n^n$ was ...
1
vote
1answer
36 views

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$

show that if $m,n\in\mathbb{N}$ are such that $(m,n)=1$, then $$\frac{(m+n-1)!}{m!n!}\in\mathbb{N}.$$ I have a theorem (shown in my text) that says $$\fbox{If $a_1,...,a_m\in\mathbb{N}^*$ then ...
5
votes
2answers
118 views

Show that there is no natural number $n$ such that $3^7$ is the largest power of $3$ dividing $n!$

Show that there is no natural number $n$ such that $7$ is the largest power $a$ of $3$ for which $3^a$ divides $n!$ After doing some research, I could not understand how to start or what to do to ...
4
votes
3answers
80 views

If $N$ is a multiple of $100$, $N!$ ends with $\left(\frac{N}4-1 \right)$ zeroes.

Did certain questions about factorials, and one of them got a reply very interesting that someone told me that it is possible to show that If $N$ is a multiple of $100$, $N!$ ends with ...
2
votes
1answer
44 views

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property?

Find the smallest value of $n$ so that the greater potency of $5$ which divides $n!$ is $5^{84}$. What are the other numbers that enjoy this property? I thought I would put together an equation ...
0
votes
2answers
40 views

For $j \in \{0,…,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true?

For $j \in \{0,...,n-1\}$ is $(n-j)!(j+1)! \leq n!$ true? I mean $\dfrac{n!}{(n-j)!(j+1)!}$ doesn't have to be an integer. I need this inequality in another exercise, so Is it provable?
8
votes
2answers
153 views

Find the greatest power of $104$ which divides $10000!$

Find the greatest power of $104$ which divides $10000!$ I thought $$104=2^3\cdot13$$ so I have to find $n$ such that $$(2^3\cdot13)^n\mid 10000!$$ Obviously, we can see that there are fewer ...
1
vote
1answer
94 views

How to derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n+ 1)$?

How can you derive $\frac{d}{dn}n! = \Gamma(1 + n)\psi^{(0)}(n + 1)$? I have tried checking Wolfram Alpha for a step-by-step solution, but none is given. Moreover, of what is the second function, ...
2
votes
1answer
150 views

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number.

Find the prime factor decomposition of $100!$ and determine how many zeros terminates the representation of that number. Actually, I know a way to solve this, but even if it is very large and ...
0
votes
1answer
105 views

Finite calculus: Apply difference operator to generalized falling factorial $(ax+b)^{\underline m}$

The $m$th falling factorial power of $x$ is defined as $x^{\underline m}:=x(x-1)...(x-m+1),$ and the difference operator as $\Delta f(x) := f(x+1)-f(x).$ One fundamental statement in finite ...
3
votes
1answer
68 views

Simplifying this logarithm series

$$\sum_{i\; =\; 2}^{99}{\frac{1}{\log _{i}\left( 99! \right)}}$$ How would you evaluate (or at least simplify) this logarithm series?
1
vote
1answer
114 views

How does Knuth's second algorithm to calculate permutations work?

I have started reading the Art of Computer Programming Volume 1 by Knuth. The first half of the book is basic concepts in maths. On page 45 there is an algorithm to obtain the next (amount of) ...
0
votes
1answer
54 views

help on manipulating this algebraic expression

So I have something like: $\frac {k!}{(k-3)!3!}$ I'm going to add $\frac 12k(k-1)$ to this, and I want to obtain $\frac {(k+1)!}{(k-2)!3!}$ as the result. I'm having trouble with this since I need ...
0
votes
4answers
218 views

A sum involving Fibonacci numbers, $\sum_{k=1}^\infty F_k/k!$

Let $F_k$ be Fibonacci numbers. I am looking for a closed form of the sum $\sum_{k=1}^\infty F_k/k!$. I tried to use Wolfram Alpha, but it is not doing the sum Fibonacci[k]/k! , k=1 to infinity. ...
0
votes
3answers
94 views

determining the greatest n in which 3^n divides 30!

Determine the greatest integer n such that $3^n\mid 30!$ I have no idea of how to approach this problem. I would first calculate $30!$ but obviously that number is way too large. Any help?
0
votes
2answers
33 views

Difference operation on factorials

Please how is the combination addition formula ${{t}\choose{r}}={{t-1}\choose{r}}+{{t-1}\choose{r-1}}$ useful in proving the difference equation $\Delta_{t}{{r+t}\choose{t}}={{r+t}\choose{t+1}}$? ...
0
votes
3answers
290 views

Why factorials when divided by factorials less than the number have a remainder 0?

Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y2!\cdots y_n!} $as long as Summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the ...
1
vote
1answer
85 views

Properties of the $\text{lcm}(1,2 ,… n)$ function

I was thinking the other day about the following function - a sort of prime factorial: $$f(n) = \text{lcm}(1,2,\cdots,n) $$ Does this function have a name? Does it have any interesting properties ...
1
vote
4answers
55 views

Is there any number which $n!$ is lower than $2^n$ or same?

I interested in this question. how many numbers meet this condition? I think a few of them meet this but I want a proof for this. also I'm not very pro in mathematics.
1
vote
1answer
53 views

how is a factorial fraction equal to the product notation

how is the $\prod$(2k-3) from k=2 to n equal to : ${(2n-3)!\over 2^{n-2}(n-2)!}$ where n $>=$ 2 i know that the (2n-3)! is equal to the product of 2k-3 from k=2 to n but I can't figure out the ...
2
votes
1answer
55 views

Solving a permutation/Combination equation

please help check if this would the correct way to solve this: $^nP_2 = ^{n+1}C_3$. I want to solve for $n$. theoretically, I was thinking that: $^nP_k = k!\times ^nC_k $ hence: ...
0
votes
1answer
61 views

Constant distribution in the factorial 2n!

a very simple question: how does 2 distribute in the factorial $2n!$ ? $2((n)(n-1)(n-2))$ would be treating the expression as $2(n!)$, which is different from: $(2n)(2n-1)(2n-2)$ which is ...
38
votes
14answers
4k 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 ...
1
vote
1answer
33 views

Homework - algebra, find constants

The question is as follows, I think I solved it partially: Show that there are $a,b$ real positive numbers such that $an^7 \leq \frac{n!}{7!(n-7)!} \leq bn^7$ $7\leq n$ my solution for b: ...
2
votes
2answers
112 views

Prove that $n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}$

Prove that for all $n \in \mathbb{N}$, $$n(n^2 - 1) = \frac{(n+1)!}{(n-2)!}.$$ Thanks in advance.
2
votes
2answers
131 views

Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$

Show that $\frac{(2n)!}{(n)!}=2^n(2n-1)!!$ is the question I am struggling with. I started by saying: $(2n)!=2n(2n-1)(2n-2)(2n-3)...3*2*1$ But then I'm stuck.
2
votes
4answers
71 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?
3
votes
1answer
54 views

I don't how do you call this, but please help me. Calculate the result of…

How to calculate the result of: $$\frac{2012(1!)}{3!}+\frac{2012(2!)}{4!}+\frac{2012(3!)}{5!}+...+\frac{2012(2010!)}{2012!}$$ What is the theory used? Sequence of number? Please help me understand ...