\begin{align*} M''(0) & = \frac{{\left(p^{2 \, r - 1} r - p^{2 \, r - 2} r - p^{2 \, r - 2}\right)} {\left(p - 1\right)} r}{p^{2 \, r}} \\ & = \left ( \frac{r(1-p)}{p^2} \right )^2 + \left ( \frac{r}{p} \right )^2. \end{align*}