For $N$ Poisson with parameter $\lambda$ and fixed $a>1$. Calculate
$$\lim\frac{P(N \geq k)}{P(N=k)},$$
with the limit taken for $\lambda\to\infty$, $k\to\infty$, $k/ \lambda \to a$. (Hint: Geometric series)
I found that
$$\frac{P(N \geq k)}{P(N=k)}=\sum_{n=k}^\infty \lambda^{n-k}\frac{k!}{n!}$$
But I couldn't figured out what kind of geometric series to use. Also since we know $1/a<1$, I thought to obtain something like $\lambda/k$ somewhere but couldn't succeed.
Thanks for any help!