A factorial is the product of all positive integers less than or equal to a given integer.

### Factorial Defined

 A factorial is the product of all positive integers less than or equal to a given integer $$n$$, and is generally written in the form $$n!$$, or by old notation, as $$\underline{\!\left| n\right.}$$. By convention, the factorial of zero is taken to to be 1. The factorial function is generally defined as:  $$n! = \sum\limits_{k=1}^{n} k$$ $$=1\cdot\ldots\cdot n$$$$(n > 0)$$ This can also be written as a recursive function:  $$n! = n\times(n-1)!$$

### Uses of Factorials

 The factorial function is especially useful in certain infinite series functions, such as the Factorial Series:  $$\epsilon = \sum\limits_{n=0}^{\infty}\frac{1}{n!}$$ $$=\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\cdots +\frac{1}{\infty!}$$ The resulting value of this series, epsilon (ε), is used in natural logarithms, and is especially useful in exponential growth and decay calculations. Factorials are also used in permutaions and combinations calculations:  $$_nP_r = \frac{n!}{(n-r)!}$$  $$_nC_r = \frac{n!}{(n-r)!r!}$$ Permutations and combinations functions are generally included in discussions on statistics and probability, and as such, will not be explained here.