\begin{align*} M(t) & = \sum_{x=1}^{\infty} e^{tx} p (1-p)^{x-1}\\ & = \frac{p}{1-p} \sum_{x=1}^{\infty} (e^t (1-p))^x\\ & = \frac{p}{1-p} \frac{e^t (1-p)}{1 - e^t (1-p)}\\ & = \frac{pe^t }{1 - e^t (1-p)}. \end{align*}