For primes $p_1$ and $p_2$ and a monotonically increasing sequence of natural numbers $a_i$, if $\sum_{i=0}^\infty p_1^{i}p_2^{-a_i}$ converges and is rational does that imply that $\sum_{i=0}^\infty p_2^{-a_i}$ is also rational?
The converse is easy to show (assuming convergence), but I can't seem to make it work in the forward direction. It's also interesting whether there are certain choices of $p_1$ and $p_2$ that make the problem easier or harder, since for example $p_2 = 2$ means that $\sum_{i=0}^\infty p_2^{a_i}$ is an arbitrary real between 0 and 1.