A convergent series $\sum r_m$ such that $\sum r_m^q$ diverges for each $q\in (0,1).$

Question

Let $\sum_{m} r_{m} < \infty$ be a convergent series such that for all $m$, $r_{m}\in (0,1)$. Can I find some rational $q \in (0,1)$ such that $\sum_{m} r_{m}^{q} < \infty$?

If no, is there a counterexample? I.e, a series $\sum_{m} r_{m} < \infty$ such that for all $m$, $r_{m}\in (0,1)$ and such that for all rational $q \in (0,1)$, $\sum_{m} r_{m}^{q} = \infty$?