I've been trying to make sense of the gamma function and why some values pop up, like $\operatorname{\Gamma}\left (\dfrac{3}{2} \right) =\dfrac{\sqrt{\pi}}{2}$
I'm still trying.
But here's a thought: what if you used the MacLaurin series for $e^{-x}$ instead. Here's what I came up with:
$n! = \displaystyle \int_0^\infty x^n \displaystyle \sum_{k=0}^\infty \dfrac{(-1)^k \cdot x^k}{k!} = \displaystyle \int_0^\infty \displaystyle \sum_{k=0}^\infty \dfrac{(-1)^k \cdot x^{k+n}}{k!}$
$=\left [\displaystyle \sum_{k=0}^\infty \dfrac{(-1)^k \cdot x^{k+n+1}}{(k+n+1) \cdot k!} \right]_0^\infty$
$= \left [\displaystyle \sum_{k=0}^\infty \dfrac{(-1)^k \cdot x^{k+n+1}}{(k+1)! + nk!} \right]_0^\infty$
Evaluating that at $0$ is just $0$, but it's the upper bound I'm worried about.
Is this reasoning:
- Open to improvement or
- Totally wrong?