Convergent sum / divergent product

Hi guys,

This is apparently a fairly well-known example, but I can't seem to figure it out. Anyone know this? Want

but

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.

The sequences

don't seem to work, although I might be wrong about this last one. what about

Re: Convergent sum / divergent product

Seems I've answered my own question as

works!