Let $\rho\in]0,1[$ and $(u_n)_{n\in\mathbb{N}}$ a sequence of positive integers such that $\lim_{n\to\infty}u_n=\infty$. Consider the following series: $$\sum_{n\in\mathbb{N}}^\infty \rho^{u_n}$$ Q : What is the condition on the sequence $(u_n)_{n\in\mathbb{N}}$ such that the above series converges ?
So far, I can see that it is enough to have $u_n<u_{n+1}$.
But does $\lim_{n\to\infty}u_n=\infty$ suffice ?
Let $a_n:=\rho^{u_n}$, which is virtually an arbitrary sequence. You have little other option than studying the convergence of $a_n$, for which there isn't a universal criterion.
For instance, with $$a_n=\log_{1/\rho}(n),$$
we have an harmonic series, for which the ratio and root tests are known to fail.