Given any increasing positive sequence $\{a_n\}$ diverging to infinity, is it possible to construct a non-negative sequence $\{b_n\}$ so that $\{b_n\}$ is summable but $\{a_n b_n\}$ is not? In other words, can we construct two series with arbitrarily small ratio growth but only one of them diverges?
(Edited) is it possible to find b_n so that {b_n} and {a_n b_n} are decreasing?
[This is not an answer to the question in the present form but it was a correct answer for the earlier version]. Choose $n_1<n_2<...$ such that $a_{n_{k}} >k$ Let $b_n=\frac 1 {ka_n}$ if $n =n_k$ and $b_n=0$ if $n$ is not of the type $n_k$. Note that $\sum a_nb_n =\sum \frac 1 k =\infty$ and $\sum b_n <\sum \frac 1 {k^{2}} <\infty $. If you want $b_n$'s to be strictly positive simply take $b_n=\frac 1 {n^{2}}$ when $n$ is not of the type $n_k$.