Suppose $(b_n)$ is an unbounded (real or complex) sequence. Does there always exist some (absolutely) convergent series $\sum a_n$ such that $\sum |a_nb_n|$ (or better, $\sum a_nb_n$) diverges? If so, do we have an easy explicit construction for it?
I think the answer is yes by the Banach-Steinhaus theorem, but I can't find an explicit construction.
We can choose a subsequence $b_{n_k}$ such that $|b_{n_k}| > 2^k.$ Define $a_{n_k} = 1/[k^2\text { sgn }(b_{n_k})],$ $a_n = 0$ for all other $n.$ Then $\sum a_n$ converges (absolutely) and the $n_k$th partial sum of $\sum_n a_nb_n$ is
$$\sum_{j=1}^{k}a_{n_j}b_{n_j} > \sum_{j=1}^{k}\frac{2^j}{j^2} \to \infty.$$