# Questions tagged [factorial]

Questions on the factorial function, $n!=n\cdot(n-1)\cdot...\cdot1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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### Evaluate $\frac {1+\frac {2^2}{2!} +\frac {2^4}{3!}+\frac {2^6}{4!} +\dots}{1+\frac {1}{2!}+\frac {2}{3!}+\frac {2^2}{4!}+\dots}$

Evaluate the given series $$\dfrac {1+\dfrac {2^2}{2!} +\dfrac {2^4}{3!}+\dfrac {2^6}{4!} +....}{1+\dfrac {1}{2!}+\dfrac {2}{3!}+\dfrac {2^2}{4!}+....}$$ If we factor out $\dfrac {1}{2^2}$ from the ...
1answer
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### Expressing the coefficients of $(1-x)^{1/4}$ using factorials

From the fact that $1\times3\times5\times\ldots\times(2n-1)=\frac{(2n)!}{2^nn!}$, we can show that $$(1-x)^{1/2}=\sum_{n=0}^\infty \frac{(2n-2)!}{(n-1)!n!2^{2n+1}}x^n.$$ However, can I do the same ...
2answers
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1answer
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### Beta function and gamma function

I would like to ask if someone could help me with following equation. \begin{equation} \Gamma(m)\,\Gamma(n) = \int_{0}^{\infty}x^{m-1}e^{-x}\,dx\,\int_{0}^{\infty}y^{n-1}e^{-y}\,dy \end{equation} \...
1answer
49 views

### For which $n$ is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$?

For how many values of $n$, is $\frac{n!}{4}$ equal to $\left\lfloor \sqrt{\frac{n!}{4}}\right\rfloor\left(\left\lfloor\sqrt{\frac{n!}{4}}\right\rfloor + 1\right)$? Further more, is there a way to ...
1answer
68 views

### Wallis integral and gamma function

I would like to ask if anyone would help me to explain how to reach the following relation. \begin{equation} \int_0^1 \left( 1-x^{\frac{1}{p}} \right)^q dx= \frac{p!\,q!}{(p+q)!} \end{equation} If we ...
0answers
32 views

### Factorial for positive fractions

I know that factorial is also represented by Gamma function and we have exact value of factorial of (1/2), can someone help me how to find numerically(approximate) the factorial of positive fractions ...
2answers
129 views

### How to calculate limit as $n$ tends to infinity of $\frac{(n+1)^{n^2+n+1}}{n! (n+2)^{n^2+1}}$?

This question stems from and old revision of this question, in which an upper bound for $n!$ was asked for. The original bound was incorrect. In fact, I want to show that the given expression ...
2answers
111 views

### Show that $n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ for $n\ge 2$

$n! < \frac{(n+1)^{n^2+n+1}}{(n+2)^{n^2+1}}$ For very small values of $n$ (i.e. $2\le n\le 6$) the function on the right nicely approximates $n!$ before significantly overtaking it. I don't have ...
1answer
80 views

### Proof by induction for nth derivative

Show the following hold by induction: $$\frac {d^n}{dx^n}\frac {e^x - 1}{x} = (-1)^n \frac{n!}{x^{n+1}} \left( e^x \left(\sum_{k=0}^{n} (-1)^{k} \frac{x^k}{k!}\right) - 1 \right)$$ Proof. It's not ...
2answers
42 views

### Simplifying factorial with different coefficient [closed]

$$\frac{(pn)!}{(qn)!},\quad p\not = q$$ If possible, how could I simplify the above factorial?
2answers
44 views

### Logarithm of factorial equal to sum of logarithm of primes

Let $N$ a positive integer. Denote $\mathcal{P}$ the set of prime numbers. I have to show that \begin{align} \log(N!) = \sum_{p^{\nu}\leq N \\ p\in \mathcal{P}} \left\lfloor\dfrac{N}{p^{\nu}}\right\...
1answer
78 views

### what is the root of this polynomial?

Let $f_n(x)=\prod\limits_{i=1}^n (x+i)-n!=(x+1)(x+2)\cdots(x+n)-n!$ $n$ is a positive integer. What are the roots of the polynomial for a given $n$ except $0$? Or determine the real part of the ...
3answers
109 views

### Why do the factorials appear in differences of consecutive powers?

Why do the factorials appear when repeatedly taking the differences of consecutive powers? Or rather why is the $n_{th}$ factorial equal to the $n_{th}$ difference of $(k+1)^{n}-k^n$? I'm having ...
0answers
35 views

### What's a compact way to express the product of terms in arithmetic progression?

There is a notation for the product of the first $n$ positive integers. My question is whether it is possible to express the product of any other non-trivial arithmetic progression with $n$ terms ...
4answers
299 views

### The asymptotic behavior of $n\ln n -n$ [closed]

How do I show that $$\displaystyle\lim_{n\to\infty}\dfrac{n\ln n - n}{\ln n!}=1?$$
1answer
29 views

### Need proof/explanation for a problem involving factorials

Suppose you have a list of integers from 1 to N And say you remove any two numbers, X and Y, from the list of numbers, and add (X + Y + X.Y) to the list If you keep doing this until you have only ...
2answers
44 views

### Product representation of $(2n)!$?

I know that $n!:=\prod_{k=1}^nk$. Would it just be $(2n)!:=\prod_{k=1}^n2k$ or $(2n)!:=\prod_{k=1}^{2n}k$ Any ideas?
1answer
49 views

### Euler Representation of the factorial

I know that Euler proved that $\displaystyle z! = \int_{0}^{1} \left(- \ln\left(t\right)\right)^z dt$ How can this be proven?
2answers
38 views

### Is there a sum of “n+j” terms equivalent to n!

Is there any function so that... $$\sum_{k=0}^{n} f(k) = n!$$ or, $$\sum_{k=0}^{n+j} f(k) = n!$$ where j is any arbitrary integer?
5answers
37 views

### Factorial quotient with same number of factors

I have this limit: $$\lim_{x\to \infty} \frac{n!(3n)!}{((2n)!)^2}$$ How can I resolve it? I know that it results $+\infty$
2answers
65 views

1answer
33 views