Random variables depending on a Poisson variable

Question

Consider a Bernoulli experiment of parameter $p$ that can be repeated independently for $N$ times, where $N \sim $ Poisson$(\lambda)$. Let $X$ be the number of successes and $Y$ the number of failures. How can I calculate the distribution of $X$ and $Y$? My attempt with $X$ was to consider it as a Binomial with parameters $N,p$ (and $Y$ can be considered as a Binomial with parameters $N,1-p$). The only thing that came out in my mind was to use the disintegration formula as it follows: $$P(X=k)=\sum_{n=k}^{\infty}P(X=k \mid N=n)P(N=n)=\sum_{n=k}^{\infty} \binom{n}{k}p^k(1-p)^{n-k}\dfrac{\lambda ^n}{n!}e^{-\lambda}=\\ =\dfrac{p^ke^{-\lambda}}{k!}\sum_{n=k}^{\infty}\dfrac{(1-p)^{n-k}}{(n-k)!}\lambda^n$$ Is this correct? And if it is, how do I evaluate the last series (Wolphram Alpha says $\lambda^ke^{\lambda-\lambda p}$)?