\[Approximation\:of\:the\:Gamma\:function\:I\:made\]

\[ \Gamma(z) = \int_{0}^{\infty} t^{z-1} e^{-t} \, dt = \lim_{k \, \to \, \infty} \sqrt{2\pi}\,(z+k)^{z-\frac{1}{2}} e^{-z-k} \! \left( \prod_{n=0}^{k-1} \frac{z+k}{z+n} \right) \]

Graph