Evaluating Factorial Using Gamma Function

Question

The gamma function is the function $\Gamma: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{\geq 0}$ defined by $$ \Gamma(t)=\int_{0}^{\infty} x^{t-1} \exp (-x) \mathrm{d} x $$ 8.1: If we use the fundamental theorem of calculus, we get $\Gamma(1)=1 .$

8.2: The gamma function has the following useful property: $$ \forall t>0, \quad \Gamma(t+1)=t \Gamma(t) $$ How could we show that $\Gamma(r)=(r-1) ! $ for every $r \in \mathbb{N}$ using Exercise 8.1 and (8.2)

Answer 1

Nothe that if $\Gamma(z+1)=z\Gamma(z)$, with $\Gamma(1)=1$ then if $z$ is a positive integer we have, $$\Gamma(n)=\Gamma((n-1)+1)=(n-1)\Gamma(n-1)=(n-1)\Gamma((n-2)+1)=\ldots =(n-1)!$$