\begin{eqnarray*} E[X] &=& \sum_{k=0}^{\infty} k \; P_k \\ &=& \sum_{k=0}^{\infty} k \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} k \frac{\lambda^k \; e^{-\lambda} }{k!} \\ &=& \sum_{k=1}^{\infty} \frac{\lambda^k \; e^{-\lambda} }{(k-1)!} \\ &=& \lambda \sum_{k=1}^{\infty} \frac{\lambda^(k-1) \; e^{-\lambda} }{(k-1)!} \\ &=& \lambda \sum_{k'=0}^{\infty} \frac{\lambda^{k'} \; e^{-\lambda} }{k'!} \quad (k':=k-1)\\ &=& \lambda \cdot 1 \\ &=& \lambda \end{eqnarray*}