Probability that an integer number having Poisson distribution is even

Probability that an integer number having Poisson distribution is even

The probability $P(X=n)$ that an event X takes place $n$ times in a fixed
period of time follows the Poisson distribution with parameter $\lambda$
i.e.
$$ P(X = n) = e^{-\lambda} \frac{\lambda ^ n}{n!}$$
I have to evaluate the probability that the event $X$ takes place an even
number of times. I know that:
$$ P(X \text{ is even} ) = e^{-\lambda} \cdot \sum_{k=0}^{+\infty}
\frac{\lambda^{2k}}{(2k)!}$$
but I can't solve the series.
I guess that I have to use the fact that $e^\lambda = \sum_{n =
0}^{+\infty} \lambda^n/n!$, but I got stuck.
How can I evaluate $P(X \text{ is even})$ (alternative solutions
appreciated).