If I have all the first $n$ moments of a discrete random variable $Z$ defined on the non-negative integers $$\mathbb{E}(Z),\mathbb{E}(Z^2),\dots,\mathbb{E}(Z^n)$$ but desire $P(Z \geq 1)$, Markov's inequality can bound this from above with $\mathbb{E}(Z)$.
Can I do better using the remaining $(n-1)$ moments at my disposal? Are there better bounds on $P(Z \geq 1)$ which are more precise when more moments are incorporated?
One can use the factorial moment bound
$$P(Z \geq 1) = 1 - \sum_{i \geq 0} \frac{(-1)^i}{i!}\mathbb{E}\left[\left( Z\right)_i\right]$$ where $\left( Z\right)_i = Z(Z-1)\dots(Z-i+1)$ is the descending factorial.