If the infinite series $\sum_{k=1}^\infty a_k$ is convergent and the sequence^{1} $(b_k)_{k\in\mathbb N}$ is bounded and monotonic, then the series $\sum_{k=1}^\infty a_kb_k$ is also convergent.
Proofs: 1
If you want to apply this convergence criterion to complexed-valued series $\sum_{k=1}^\infty a_kb_k$, you have to make sure that $(b_k)_{k\in\mathbb N}$ remains a real-valued sequence. Otherwise, the monotonicity criterion will lose its sense. ↩