[Math] Taylor or Maclaurin series for the factorial function

Question

I am new to Taylor/Maclaurin series and want to know if there is a series representation for the factorial function?

Answer 1

If you mean the standard factorial function $!:\mathbb{N}\rightarrow\mathbb{N}$, a Taylor series cannot exist because $\mathbb{N}$ has no accumulation points so you cannot take derivatives of it. To see this, you could try evaluating the following difference quotient:

$$\lim_{x\rightarrow n}\frac{x!-n!}{x-n}.$$

Immediately you can see this limit is not well-posed because the factorial only takes integer values. You can't have $1.00001!$ for instance so you can't get arbitrary close to $n$.

You need some way to extend the idea of a factorial to the real numbers in order to take derivatives. One such generalization of the factorial to (almost all) real numbers is the Gamma function. For natural numbers, we have that $\Gamma(n+1) = n!$ and you can show this pretty easily. In this case, derivatives of all orders exist and you can have a Taylor series by evaluating around a certain point (that is not $0$ or a negative integer since the Gamma function is not defined for them).