The function 
$$
f : z \in \mathbb{C} \longmapsto \sum_{i} \frac{a_i}{1-a_iz}
$$
is meromorphic on $\mathbb{C}$ and has integral Taylor coefficients. It follows from a theorem of Borel that such a function must be in $\mathbb{Q}(z)$; see for example Richard Stanley's answer [here][1]. In particular $f$ has only finitely many poles ; this implies that only finitely many of the $a_i$'s are nonzero.


  [1]: https://mathoverflow.net/questions/6179/does-there-exist-a-meromorphic-function-all-of-whose-taylor-coefficients-are-pri