Seems I've answered my own question as
This is apparently a fairly well-known example, but I can't seem to figure it out. Anyone know this? Want
Now, we know that the sum will have to be conditionally convergent, as
and moreover, it is a theorem that if the sequence is square-summable, i.e.
then the product and the sum (without absolute values) converge and diverge together.
don't seem to work, although I might be wrong about this last one. what about