Counterexample for numerical series

Question

Is that possible to have two diverging series $\sum u_n$ and $\sum v_n$ where $\forall n \in \mathbb N$, $u_n v_n \geqslant 0$ and $\sum \sqrt{u_n v_n}$ converges ?

Thank you for your answer.

Answer 1

A simple and obvious example would go as follows: at odd $n$, let $u_n={1\over n}$ and $v_n={1\over n^3}$, and at even $n$ switch them the other way around: $u_n={1\over n^3}, v_n={1\over n}$.

The question would go up to an entirely new level of difficulty if it were asking about monotonically decreasing $u_n$ and $v_n$. Still, the answer is possible even then.