Exponential Series

\[e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\dots=\sum_{k=0}^{\infty}{\frac{x^k}{k!}}\]

Evaluate \(\sum_{k=0}^{\infty}{\frac{\lambda^ke^{-\lambda}}{k!}}\)

\[ \begin{aligned} \sum_{k=0}^{\infty}{\frac{\lambda^ke^{-\lambda}}{k!}}&=e^{-\lambda}\sum_{k=0}^{\infty}{\frac{\lambda^k}{k!}} \\ &=e^{-\lambda}(e^\lambda) \\ &=e^{-\lambda+\lambda} \\ &=e^0 \\ &=1 \end{aligned} \]