Prove that this is a legitimate PMF

Question

I know that these are two properties of PMF.

Non-negativity

Sum over the support equals 1

However I can't show that this PMF's sum over the support equals 1.

Answer 1

Using the Taylor series for $e^{\lambda}$ you have $$ \sum_{k=0}^{\infty} p_X(k) = \sum_{k=0}^{\infty} e^{-\lambda} \frac{{\lambda}^k}{k!} = e^{-\lambda} \sum_{k=0}^{\infty} \frac{{\lambda}^k}{k!} = e^{-\lambda} e^{\lambda} = 1 $$