Can anyone think of sequences $\{a_n\}$, $\{b_n\}$ such that $\sum a_n$ diverges, ${b_n}\to\infty$, but $\sum a_nb_n $ converges?
Thank you.
Note that $\{a_n\}$ must have infinitely many positive terms and infinitely many negative terms.
Edit: I get the feeling that only Qiaochu Yuan could answer this one... ;)