Hello,

I'm trying to teach myself the Gamma Function from here:

The Gamma Function

However, I'm a tad confused at some parts.

I don't understand at all how they figured out (derived) the inequality where it says

"A careful analysis of the Gamma function (especially if we notice that is a convex function) yields the inequality"

...and I don't know what a convex function is. I don't really see the significance in deriving the function in all these different ways.

Also, when they give the intial formula, I don't understand how it was derived/where it came from. Does anyone know how it was formulated?

$\displaystyle \int^{\infty}_0 e^{-t}t^n\,dt=n!$

Thanks for all your help.