Let $\sum_{m=1}^{\infty} n_{m} r_{m} < \infty$ be a convergent series such that for all $m$, we have that $r_{m}\in (0,1)$ and $n_m$ is a natural number. Can I find some rational $q \in (0,1)$ such that $n_{m}r_{m}^{q} \rightarrow 0$ as $m\rightarrow \infty$?
If no, is there a counterexample? I.e, a series $\sum_{m=1}^{\infty} n_{m} r_{m} < \infty$ such that for all $m$,we have that $r_{m}\in (0,1)$ and $n_m$ is a natural number, such that for all rational $q \in (0,1)$, $n_{m}r_{m}^{q} \rightarrow 0$ does not hold?
Try $n_m = 2^m$ and $r_m = 2^{-m}/m^2$.