Let $X$ be a random variable that has a Poisson distribution, with parameter $\beta$. Let $\mathbb{E}$ denote its expectation. Let $\alpha>0$ be an integer. How fast does the function of $\beta$,
$$ f_\alpha(\beta) := \mathbb{E} \bigg[ \frac{1}{X^\alpha} \boldsymbol{1}_{[X>0]} \bigg] $$
goes to zero as $\beta$ goes to infinity? Is it the case that $f_\alpha(\beta) \sim \beta^{-\gamma}$ for some $\gamma>0$ which depends on $\alpha$? Or is the speed of convergence faster than polynomial?