Suppose $X \sim Pois(\lambda)$. I'm interested in an upper bound on the ratio, $$\dfrac{P(X \leq n)}{P(X \leq n-1)}\,,\,\,\text{for n=1,2,3,...}$$ Observe that, the ratio is $>1$ & as $n \to \infty,\,$ the ratio $\to 1$. Thus, It is interesting to see if there exists some constant $K>1$ (depending on whether $\lambda \leq n$ or $\lambda \geq n$) so that the ratio is $\leq K^{1/n}.$

Any comments on this?