Let $\Gamma$ be the pointwise limit on $]0, +\infty[$ of the sequence $\left(\Gamma_n\right)_{n \geqslant 1}$ where
$$\Gamma_n(x) = \frac{1}{x} e^{-\gamma x} \prod_{k=1}^{n} \frac{e^{x k^{-1}}}{1 + x k^{-1}}.$$
Show that the function $\Gamma$ is of class $\mathscr{C}^2$ and that, for all $x \in ]0, +\infty[$,
$$(\ln(\Gamma))''(x) = \sum_{k=0}^{+\infty} \frac{1}{(x+k)^2}.$$