No, you don't know that both and converge. Take and .
I don't think it is possible for both to diverge. This is clear if one of the sequences has only positive terms (or similarly only negative terms). For the case with mixed terms try violating a condition for the alternating test. I haven't worked out all the details but think that might work. I'll let you know I have figured out more.
From the wonderful book Counterexamples in analysis, an example of two (very!) divergent series whose Cauchy product is absolutely convergent. Define , for n≥1. Define , for n≥1. Then it's easy to check that but for n≥1.