\begin{equation*} M(0) = 1 \end{equation*}

\begin{equation*} M'(0) = \frac{1}{p} = \mu \end{equation*}

\begin{equation*} M''(0) = \frac{1-p}{p^2} + \frac{1}{p^2} = \sigma^2 + \mu^2 \end{equation*}