As a counterexample -- while looking for two divergent sequences whose sum or product converges we have the obvious choice, (1, -1, 1, -1, . . . ) with (-1, 1, -1, 1 . . . ) but is there another cool example of this?
Edit: disregard this. The left forum sidebar protruded over the formula for and obscured the minus.
Isn't the harmonic series? Then it does not converge.