> [!NOTE] Theorem > Let $\Gamma$ denote [[Gamma Function|gamma function]]. For all $n\in \mathbb{N},$ $\Gamma(n+1)=n!,$ the $n$th [[Factorial|factorial]]. **Proof**: This proof assumes [[Integral Form of Gamma Function]]. By [[Integration by Parts]], for all $n\in \mathbb{N},$ we have $\int_{0}^{m} e^{-x} x^{n} \, dx = [-e^{-x}x^{n}]_{0}^{m} + \int_{0}^{m } e^{-x} n x^{n-1} \, dx $Since $m^{n}e^{-m}\to 0$ as $m\to \infty,$ we have $\int_{0}^{\infty} e^{-x} x^{n} \, dx = n \int_{0}^{\infty} e^{-x} x^{n-1} \, dx $that is, $\Gamma(n)=n \Gamma(n-1).$ Also $\Gamma(1)=\int_{0}^{\infty} e^{-x} \, dx=1.$ Thus for all $n\in \mathbb{N},$ $\Gamma(n+1)=n!.$