Factorial

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In mathematics, factorial, can be called fucktorial from an unknown loser is a unary operator on the set of natural numbers. Go give n any numbers, but n can't be less than 0, the number must not too big as hell like Rayo's number! The "n factorial", denoted by $n!$ by an idea from the chirping sound, as the product of the first n positive natural numbers. It's like:

$$n!=1\times 2\times 3\times 4\times \dots \times n$$

Or to make people understand it easier we could also write it as

$$n!=n\times (n-1)\times (n-2)\dots \times 3\times 2\times 1$$

Especially n is NOT defined when it is a negative integrals and if n were 0 we must agree that $0!=1$ following the empty product rule. Or we can propose against the rule that it is undefined by using the formula $n!=n(n-1)!$. Look! Someone else is churning the mathematical law by proving that it is 0 lmao.

$$0!=0\times (0-1)!=0\times (-1)!=0$$

About the surprise exclamation mark notation it is used by a christian named Kramp so it with a number is like "OMG!!!!!!!1", and it sounds like: "Chirp, chirp!" a sound created by birds every morning. The factorial is popular in many different areas of mathematics and yes it is also very useful in geometry, and most notably combinatorics, since it is the number of different ways to shuffle a group of some n objects, of course in combination variant the formula is like: ${_{n}}C_{k}={\frac {n!}{k!(n-k)!}}$ (with $0<k\leq n$ condition)

Stirling approximation

An approximation form of the factorial proposed for calculating real and very big numbers. It is a good and a highly basic formula that lead the accurate results even if n has small value, but it is useless, can't even calculate negative rational numbers. Named after James Stirling, although it is Moivre's discovery and James might have stolen it.

$$n!\sim {\sqrt {2\pi n}}\cdot e^{-n}n^{n}$$

Double factorial

We can consider the $n!!$ as the product of the first n elements of the arithmetic progression with the first element equal to 1 and the common difference equal to 1. Expanding with the common difference equal to 2 we have:

$$n!!=\left\{{\begin{matrix}1\qquad \qquad \ &&\\n(n-2)!!&&\qquad \qquad \end{matrix}}\right.$$

It is the product of the first n elements of the sequence with the first element 1 and the common difference is 2. And for even integers, the formula is like:

$$(2n)!!=\prod _{k=1}^{n}2k$$

Double factorial of egg for example:

$$8!!=\prod _{k=1}^{4}2k=384$$

Not like the dumb factorial, double factorial can have valid value of some odd negative integers not even negative integers if even negative integers then it is undefined. An overview about the special formula of double factorial:

$(n-2)!!={\frac {n!!}{n}}$

This is what it like, it is so basic that no one cares about.

Infinity factorial

We have a quick question here, what about this?

$$\lim _{x\to \infty }x!=?$$

Well this is such a big number "bigger than the number of Rayo", the answer seems easy, is infinity, right? But that is what an idiot is thinking. Be hold, the superior Riemann zeta function, the function that break every laws of mathematics, it is also unbreakable too, by confirm that this series is equal to a very weird number that isn't right in nature.

Ha! There here it, the factorial of infinity, it is only greater than 2!!

$$ln({\infty !})=ln(1)+ln(2)+\dots +ln(\infty )=-\zeta '(0)$$

Set the series is $\zeta '(0)$ but this is quite damn hard and must needed some other functions for help, this such as: $\eta '(s)=ln(2)-ln(3)+ln(4)-\dots =1-2^{1-s}$, wow an another series.

This brat needs more expanding, now.

$$\eta '(0)=\zeta '(0)+{\frac {-ln(4)}{2}}={\frac {1}{2}}\times ln\left({\frac {\pi }{2}}\right)$$

So we can prove $e^{\zeta '(0)}$ is $e^{\sqrt {2\pi }}$, it's all done. After the proof, now all we do is to pretend ur own surprising moment, to welcome this superior number, then prove it otherwise.