Let $\{a_n\}$ be a fixed non-negative sequence. Assume that the series $$\sum_{n=1}^{\infty} n^q a_n$$ converges for a $q$ that we can choose as large as needed.
Fix $p\ge 1$. Is it possible to choose $q$ in the condition above (large in relation to $p$) in such a way that $$\sum_{n=1}^{\infty} n^p\sqrt{a_n}$$ converges as well?