Everyone knows that (when $x<1$) $\sum_{n=1}^\infty x^n = \frac{x}{1-x}$.
I was wondering if one can find a simple expression (i.e. without the sum) of $\sum_{n \in \langle p_1,...,p_k\rangle} x^{n}$ where $\langle p_1,...,p_k\rangle$ denotes the set of all integers whose prime divisors are only those $k$ primes.
I considered the simplest example: $\langle p\rangle$ and that already made me stuck: $\sum_{n \in \langle p\rangle} x^{n} = \sum_{m=0}^{\infty} x^{p^m}= ???$.
**I should say that I just came up with this question and that the $\langle \rangle$ sign was made up for the sake of the question and is not a standard writing.