> [!NOTE] Theorem > The definitions of the [[Gamma Function|gamma function]] in [[Integral Form of Gamma Function]] and [[Euler Form of Gamma Function]] are equivalent: that is, for each $s>0,$ $\Gamma(s)=\int e^{-x} x^{s-1} \, dx = \lim_{ n \to \infty } \left( \frac{n^{s}n!}{s(s+1)\dots(s+n)} \right) .$ **Proof**: Let $s>0.$ STS $\frac{n^{s}n!}{\Gamma(s)s(s+1)\dots(s+n)}\to 1\tag{1}$as $n\to \infty.$ By [[Product Formula for Gamma Function]], $\Gamma(s+1)=s\Gamma(s).$ Thus $(1)$ is equivalent to $\frac{\Gamma(s+n+1)}{n^{s}n!}\to 1.$That is, we want that $\int_{0}^{\infty} \frac{e^{-x}x^{n}}{n!} \left( \frac{x}{n} \right)^{s} \, dx \to 1.$ Let $x=n u$