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
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2answers
49 views

Find the antiderivative of a function with a finite series and factorials

If $n\in\mathbb{N},s\leq n$, I know that $$ \int_0^1 t^s(1-t)^{n-s-1}dt=\frac{s!(n-s-1)!}{n!}. $$ I would like to find a similar formula: is there a function $f(t)$ such that $$ \int_0^1 f(t) ...
1
vote
2answers
94 views

Prove $ \sum_{1\leq k < n} k^{\underline{m}}=\frac{n^{\underline{m+1}}}{m+1} $ by induction on $m$

I want to prove by induction the following sum: $$ \sum_{1\leq k < n} k^{\underline{m}}=\frac{n^{\underline{m+1}}}{m+1} $$ but induction should be on $m$. Any hint will be helpful. EDIT: $ ...
3
votes
3answers
248 views

Proving that $n!≤((n+1)/2)^n$ by induction

I'm new to inequalities in mathematical induction and don't know how to proceed further. So far I was able to do this: $V(1): 1≤1 \text{ true}$ $V(n): n!≤((n+1)/2)^n$ $V(n+1): ...
4
votes
3answers
73 views

Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that$\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
6
votes
1answer
1k views

Is the sum of factorials of first $n$ natural numbers ever a perfect cube?

If $S_n = 1! + 2! + 3! + \dots + n!$, is there any term in $S_n$ which is a perfect cube or out of $S_1$, $S_2$, $S_3$, $\dots S_n$ is there any term which is a perfect cube, where $n$ is any natural ...
0
votes
1answer
102 views

Summation of reciprocal of Product of Factorials.

How can this summation be evaluated: $${∑ {1\over {a_1!a_2!....a_m!}}}$$ Where $$a_1+a_2+.....+a_m=n$$ Also $a_i !=n $ and $m<n$.
1
vote
1answer
68 views

Canceling factorials and exponentials in sum

I'm trying to understand the following the proof. I want to show that $$E\left[\frac{1}{X+1}\right] = \frac{1}{(n+1)p}(1-(1-p)^{n+1})$$ The proof goes like this: $$ \begin{align} ...
0
votes
1answer
58 views

Is this an accurate way to represent n! using Π?

I recently learned of the $\Pi$ symbol, and was wondering if the following is an accurate way to represent $n!$: $\Pi_{i=0}^{n-1} n - i$
1
vote
2answers
122 views

How to estimate the size of a ratio with very large factorials?

I want to estimate the size of the following ratio: $$\frac{10^{18}!}{10^{14}!\ 10^4!}$$ Since I don't have an idea how to simplify it and no CAS is able to handle numbers of this size, I am at an ...
5
votes
2answers
239 views

What is the largest n for n!< 1000?

This is simple factorial equation question. How do you find the largest n satisfying n! < 1000? (Edit) Actually, I want to find some other logic other than brute force. For example, How about ...
3
votes
2answers
2k 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
3answers
165 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 = ...
2
votes
1answer
358 views

How do you solve an inequality with the factorial of a variable?

How do you solve an inequality with the factorial of a variable? Example: Determine the interval of $n \in \Bbb N$ for which the following inequality holds: $$n! \leq 157788 \cdot 10^{10} $$ Can ...
1
vote
2answers
54 views

How does factorial result in the computation of possible orderings?

For 3 characters to find the ways of putting them in order: ABC, ACB, BAC, BCA, CAB, CBA $= 3! = 3\times2\times1 = 6$ When trying to find ways to put an object in order ...
0
votes
1answer
94 views

Theory behind multiplication & combinations?

If with the Binomial Coefficient we try to find the possible combinations $\binom{n}{k}$ where $n$ is equal to $k$ what is the theory behind factorial resulting in the correct solution? E.g. ...
1
vote
1answer
47 views

Prove that $\frac{n!}{(n-k)!} = n^{\underline{k}}$

I'm having some trouble proving the relation $$\frac{n!}{(n-k)!} = n^{\underline{k}}$$ Do you have to get into using gamma functions in order to prove this rigorously? Also, wikipedia seems to ...
2
votes
1answer
72 views

Counting permutations, with additional restrictions

There are 10 slots and some marbles: 5 red, 3 blue, 2 green, how many ways can you fit those marbles into those slots? Those marbles fit in 10!/(5! 3! 2!) ways ...
9
votes
1answer
291 views

How is it possible that $\infty!=\sqrt{2\pi}$?

I read from here that: $$\infty!=\sqrt{2\pi}$$ How is this possible ? $$\infty!=1\times2\times3\times4\times5\times\ldots$$ But \begin{align} 1&=1\\ 1\times2&=2\\ 1\times2\times3&=6\\ ...
0
votes
4answers
149 views

Show that the series $\sum_{n,m=1}^\infty 1/(n+m)!$ is absolutely convergent and find its sum

Show that the series $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!}$$ is absolutely convergent and find its sum. This comes from a chapter called interchange of limit operations. I tried using the ratio ...
2
votes
2answers
69 views

Divisibility and factorial

If $n = st$ and $s > 0$ and $t > 0$ then prove that $(s!)^t|n!$ . If I replace $n!$ with $ (st)!$ how can I simplify it so that I can show that the division is an integer.
2
votes
2answers
84 views

Inequality $C\lceil\log{n}\rceil! \geq n^k$

I've been struggling to prove there exist $C$ for $n, n_{0}, \forall k >0 \in \mathbb{R}$ such that $\forall n > n_{0}$: \begin{equation}C\lceil\log{n}\rceil! \geq n^k\end{equation} As you ...
1
vote
2answers
75 views

How $\alpha(\alpha+1)\ldots(\alpha+k-1)=\frac{\Gamma(\alpha+k)}{\Gamma(\alpha)}$?

Probability function of Negative Binomial Distribution, $NB(\alpha,p)$, is $$P(X=k)=\binom{\alpha+k-1}{k}(1-p)^{\alpha}p^k,\quad \alpha>0$$ Probability generating function of Negative Binomial ...
2
votes
2answers
447 views

Simple Combinatorics Problem

I've 'indirectly' studied combinatorics earlier in probability courses, but now it's part of the math course I'm taking. I always thought it was very hard, and well, here I am again... The problem ...
23
votes
5answers
473 views

How to find $\lim_{n\to\infty}\frac{1!+2!+\cdots+n!}{n!}$?

How to evaluate the following limit? $$\lim_{n\to\infty}\dfrac{1!+2!+\cdots+n!}{n!}$$ For this problem I have two methods. But I'd like to know if there are better methods. My solution 1: Using ...
1
vote
2answers
220 views

Factorial inequality.

I have the following factorial function $(m-k-1)!(k-1)!$ for $m \in \mathbb{N}$, $m \geq 2$ and $k \in \{1,\cdots,m-1\}$. I'm trying to find the value of $k$ for which the above expression attains ...
22
votes
4answers
1k views

Solutions for x!/y!=(y+1)!

I was watching a video recently, and I saw how 10*9*8*7 was equal to 7*6*5*4*3*2*1, or to make it clearer, 10!/6!=7!. I was wondering if there were any other solutions, so I checked the web, to find ...
0
votes
3answers
43 views

Simlifying [(k+1)! - 1] + (k+1)((k + 1)!)

I'm afraid I've gotten a bit rusty on Math since I was last in university. I was looking at a problem in my text book that simplified ...
0
votes
1answer
33 views

Simplify the following problem

How $$\frac{1}{k}\sum_{m=r}^{k-1}\frac{m!}{(m-r)!}=\frac{(k-1)(k-2)\ldots(k-r)}{r+1};\quad r=1,2,\ldots$$ I have thought in two ways: ...
0
votes
1answer
21 views

Simple Calculation

How $$\sum_{m=r}^{k-1}m(m-1)\ldots(m-r+1)=\sum_{m=r}^{k-1}\frac{m!}{(m-r)!};\quad r=1,2,\ldots$$ ? i have ...
3
votes
1answer
126 views

The number of zeros in the decimal representation of the factorial of 126

How many zeros are in $126!$ ... the result is $34$. But can I calculate it manually? I have seen How many zeroes are in 100! but I don't think it's helpful.
1
vote
3answers
81 views

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$.

Show that $\frac{1}{r!}-\frac{1}{(r+1)!}\equiv\frac{r}{(r+1)!}$. I get $$\frac{1}{r!}-\frac{1}{(r+1)!}=\frac{(r+1)-r!}{r!(r+1)!}$$ and in the numerator since $$(r+1)!-r!=r$$ so ...
2
votes
2answers
2k views

How many ways to split 5 numbers in 2 groups?

How many ways can you split the numbers 1 to 5 into two groups of varying size? For example: '1 and 2,3,4,5' or '1,2 and 3,4,5' or '1,2,3 and 4,5'. How many combinations are there like this? What is ...
1
vote
0answers
44 views

What are the best and most elementary bounds for $n!$?

What this question is looking for is bounds on $n!$ that are elementary in nature (I seem to have a fetish for these type of proofs). In general, as the results become more complicated, they also ...
6
votes
1answer
542 views

Solving equation involving factorial

I have the following problem: $$ N^n(N-n)!=A $$ Where $N$ and $A$ are constants. I want to solve this equation for $n$ (for a variation of the birthday problem), but I have little experience with ...
2
votes
2answers
142 views

Is it easier to calculate a factorial or the inverse of a factorial (1/n!) for extremely large n?

I need to calculate extremely large factorials but they grow extremely fast! I was wondering if it might be easier to calculate $\frac{1}{n!}$ rather than n! itself because as n goes to infinity ...
-1
votes
3answers
123 views

Prove that $n! \geq 2^{n-1}$ for $ n\geq1$ [duplicate]

Mathematical Induction:-Prove that $n! \geq 2^{(n-1)}$ for $n\geq 1$. I tried mathematical induction but could not
1
vote
4answers
135 views

How many values of $n<50$ exist that satisfy$ (n-1)! \ne kn$ where n,k are natural numbers?

How many natural numbers less than 50 exist that satisfy $ (n-1)! \ne kn$ where n,k are natural numbers and $n \lt 50$ ? when n=1 $0!=1*1$ when n=2 $1!\ne2*1$ ... ... ... when n=49 ...
0
votes
2answers
112 views

Can 44 be a factor of $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)$ where n is a natural number greater than 3?

Can 44 be a factor of $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)$ where n is a natural number greater than 3 ? When $n =4 $, $(n-3)(n-2)(n-1)n(n+1)(n+2)(n+3)=1\cdot2\cdot3\cdot4\cdot5\cdot6\cdot7$
2
votes
0answers
98 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
1
vote
1answer
542 views

How to calculate the number of permutations and combinations if k is equal to n?

Say the question is How many unique ways are there to arrange the letters in the word FANCY? The formula I use for permutations is n! / (n - k)! ...
4
votes
2answers
113 views

Approximation of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $

I would like to find an approximation when $ n \rightarrow\infty$ of $ \frac{n!}{(n-2x)!}(n-1)^{-2x} $. Using Stirling formula, I obtain $$e^{\frac{-4x^2+x}{n}}. $$ The result doesn't seem right! ...
2
votes
1answer
76 views

Limit of a function with factorial [duplicate]

$\lim_{n \rightarrow \infty} (x^n/n!)=0$. prove. x is finite whereas n is infinite. But increasing n means also increasing $x^n$. It is understandable that if n is too large n! will exceed $x^n$. How ...
3
votes
2answers
541 views

German tank problem, simple derivation [duplicate]

I was reading the recent question on the German tank problem, and had trouble with one of the derivations in this section. $$\sum_{m=k}^N m \frac{\binom{m-1}{k-1}}{\binom N k} = ...
5
votes
3answers
232 views

How Many Ways to Build a 6-Pack

There is a beverage company here that claims to have a selection of 200 different beers. They have a special deal where you can build your own six pack at a discount. They advertise that there are ...
1
vote
1answer
73 views

How many digits are 9 at the end?

How many digits at the rightmost of this sum are 9? $$1! + 2\times2! + 3\times3! +\dots +48\times48!$$ I tried to calculate the first few terms but I couldn't solve it. The answer is 10.
2
votes
1answer
7k views

How do we calculate factorials for numbers with decimal places? [duplicate]

I was playing with my calculator when I tried $1.5!$. It came out to be $1.32934038817$. Now my question is that isn't factorial for natural numbers only? Like $2!$ is $2\times1$, but how do we ...
1
vote
0answers
76 views

Limit of the sequence $\frac{(n!)^{1/n}}{n}$ [duplicate]

Which is the limit of the fllowing sequence $$\frac{(n!)^{1/n}}{n}$$
2
votes
2answers
314 views

Limit, factorials

There is the following limit, I would like to calculate: $\lim_{n\rightarrow\infty}\frac{n!}{\left(n+1/6\right)!}$ I tried to use the Stirling approaximation formula $n!\approx\sqrt{2\pi ...
5
votes
4answers
525 views

Find the sum of series $\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}$

I wanted to know how can I start to find the sum of the series: $$\sum_{n=0}^\infty\frac{(4n)!}{(4n+4)!}=\frac{1}{4!}+\frac{4!}{8!}+\frac{8!}{12!}\cdots$$ I am having no clue. Thanks.
14
votes
13answers
3k views

Can the factorial function be written as a sum?

I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.