I am looking for a proof of the following fact for reference
$$ \lim _{n\to \infty }{\frac {\Gamma (n+\alpha )}{\Gamma (n)n^{\alpha }}}=1,\qquad \alpha \in \mathbb {C} $$
Where $\Gamma$ is the Gamma function. I would like to follow the entire proof so I am hoping it is not a multipager.
Take logarithms $$\log\left({\frac {\Gamma (n+\alpha )}{\Gamma (n)n^{\alpha }}}\right)=\log (\Gamma (n+\alpha ))-\log (\Gamma (n))-\alpha \log (n)$$ and use Stirling approximation for large $p$ $$\log(\Gamma(p))=p (\log (p)-1)+\frac{1}{2} \left(\log \left(\frac{1}{p}\right)+\log (2 \pi )\right)+\frac{1}{12 p}+O\left(\frac{1}{p^3}\right)$$ You should end with $$\log\left({\frac {\Gamma (n+\alpha )}{\Gamma (n)n^{\alpha }}}\right)=\frac{\alpha(\alpha-1)}{2n}+O\left(\frac 1 {n^2}\right)$$
