\[\begin{align*} E(X) &= \sum_{k=0}^{\infty} k \cdot \frac{\lambda^k e^{-\lambda}}{k!}\\ &= e^{-\lambda} \sum_{k=1}^{\infty} k \cdot \frac{\lambda^k}{k!}\\ &= e^{-\lambda} \sum_{k=1}^{\infty} \frac{\lambda^k}{(k-1)!}\\ &= e^{-\lambda} \lambda \sum_{j=0}^{\infty} \frac{\lambda^j}{j!} \quad (j = k-1)\\ &= e^{-\lambda} \lambda e^{\lambda} \\ &= \lambda \end{align*}\]
This is a Taylor series, hence: \[ \sum_{j=0}^{\infty} \frac{\lambda^j}{j!} = e^{\lambda} \]
\[\begin{align*} E(X^2) &= E(X(X-1)) + E(X)\\ &= \sum_{k=0}^{\infty} k(k-1) \frac{\lambda^k e^{-\lambda}}{k!} + \lambda\\ &= e^{-\lambda} \sum_{k=2}^{\infty} \frac{\lambda^k}{(k-2)!} + \lambda\\ &= e^{-\lambda} \lambda^2 e^{\lambda} + \lambda \\ &= \lambda^2 + \lambda\\ Var(X) &= \lambda^2 + \lambda - \lambda^2 \\ &= \lambda \end{align*}\]
\[\begin{align*} M_X(t) &= E[e^{tX}] \\ &= \sum_{k=0}^{\infty} e^{tk} \frac{\lambda^k e^{-\lambda}}{k!}\\ &= e^{-\lambda} \sum_{k=0}^{\infty} \frac{(\lambda e^t)^k}{k!}\\ &= e^{-\lambda} e^{\lambda e^t} \\ &= e^{\lambda(e^t - 1)} \end{align*}\]