Comparing An*Bn and An+Bn in the Same Limits

[Dell]

given two series,
An and Bn, where one converges and one doesn't in the same limits

what can be said about
(An*Bn)
and (An+Bn)
in those limits

for (An+Bn) i said that since$$\sum$$(An+Bn)=$$\sum$$An+$$\sum$$Bn
so since one is infinite the sum must also be

but for (An*Bn) i don't think the same logic works.
what can be said about it?

 

[Gib Z]

Well, nothing. It can either converge, or diverge, it depends on what A_n and B_n are. Try thinking of a pair of functions where one converges and the other diverges, but where their product converges, and another pair where it diverges.