Is $$ \sum_{n=0}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z) ?$$

In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient conditions for $\sum a_nz^n$ to be in $\mathbb{C}(z)$ with all poles simple?