Can Ʃ (anbn) be divergent if Ʃan and Ʃbn are both convergent?

[Ryuzaki]

Homework Statement

Show by example that if Ʃa_n and Ʃb_n are two convergent series, then Ʃ (a_nb_n) may be a divergent series.

Homework Equations

N/A

The Attempt at a Solution

I've been trying long and hard to find an example for this particular case, but I'm unable to find one. I've found examples where Ʃa_n and Ʃb_n are divergent, but Ʃ(a_nb_n) is convergent. Can anyone please give an example? Thanks.

 

[clamtrox]

I think it's true that if Ʃa_n and Ʃb_n are absolutely convergent, then also Ʃa_nb_n converges. So maybe consider series which are convergent, but not absolutely convergent.