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.
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