Gumbel limit for gamma maxima

For the Gamma distribution with fixed positive integer shape $m$ and rate one, the survival function is $e^{-t}{\sum }_{j=0}^{m-1}{t}^j/j!$. Its sample maximum centered by $\log n+(m-1)\log \log n-\log ((m-1)!)$ has convergence in distribution to the standard Gumbel distribution, without an additional scale factor.