# Convergence rate about a limit concerning the Poisson CDF

The CDF of a Poisson distribution with rate parameter $$\lambda$$ is $$P(n;\lambda)=\sum_{k=0}^n \frac{\lambda^ke^{-\lambda}}{k!}.$$ As $$n$$ goes to infinity, the CDF would certainly approach 1.

Now, consider the case when the rate parameter is $$\xi n$$ with $$\xi\in(0,1)$$ being a given constant. According to the last answer in https://math.stackexchange.com/questions/160248/evaluating-lim-limits-n-to-infty-e-n-sum-limits-k-0n-fracnkk, the following limit should be 1: $$\lim_{n\to\infty}P(n;\xi n)=\lim_{n\to\infty} \sum_{k=0}^n\frac{(n\xi)^ke^{-n\xi}}{k!}=1.$$ Is it possible to check the convergence rate of the above limit? More specifically, I wonder whether the convergence rate is faster than $$n$$, i.e., whether $$n[1-P(n;\xi n)]=O(1)$$?

I checked using software that the convergence rate is faster than $$n$$, but I don't know how to show it rigorously. Can anyone provide some hints and insights? Thanks!