Poisson random variable power series

Question

Suppose X is a Poisson λ random variable.
(1) What is P(X is even)?
(2) By manipulating power series, compute $$ E [X(X − 1)(X − 2)· · ·(X + 1 − l)] $$ for each $l$ = 1, 2, 3,....

For (1), using taylor series expansion of $e^λ$.I got

$$ P(x=even) = (1/2)(1+e^{-2λ}) $$

I think the power series I need to use is the expansion of $e^λ$, which is $$ \sum_{k=0}^\infty e^{λ}/k! $$

but I don't know how to use this to calculate (2)

Answer 1

$$ \begin{align} E[X(X-1)...(X-(l-1))] &= \sum_{k=l}^{\infty} k(k-1)...(k-(l-1))\ P(X=k) \\ &= \sum_{k=l}^{\infty} \frac{k!}{(k-l)!} \frac{\lambda^k.e^{-\lambda}}{k!} \\ &= e^{-\lambda}\sum_{k=l}^{\infty}\frac{\lambda^k}{(k-l)!}\\ &= e^{-\lambda}.\lambda^l.\sum_{k'=0}^\infty \frac{\lambda^{k'}}{k'!} \\ &= \lambda^l \end{align} $$

Answer 2

1 seems to be correct.

For the second i would use the probability generating function and use the property that its lth derivarive evaluated at 1 is $E[X(X-1)...(X+1-l)]$.