How to multiply this series: $$(\sum_{t=\infty}^{\infty}a_{t})(\sum_{k=\infty}^{\infty}b_{k})$$

closed as offtopic by Ricardo Andrade, Jack Huizenga, Lucia, David White, Daniel Moskovich Dec 1 '13 at 6:17
This question appears to be offtopic. The users who voted to close gave this specific reason:
 "This question does not appear to be about research level mathematics within the scope defined in the help center." – Ricardo Andrade, Jack Huizenga, Lucia, David White, Daniel Moskovich
It's not a problem to multiply the series: the product is $\sum_{(t,k)\in\mathbb Z^2} a_tb_k$. The question is how to sum the double series that we have. For series with nonnegative terms summation is not a problem either: we take the supremum of all finite sums. And since any finite sum is contained in a sufficiently large square, it follows that $\sum_{(t,k)\in\mathbb Z^2} a_tb_k$ is finite whenever $\sum_{t\in\mathbb Z} a_t$ and $\sum_{k\in\mathbb Z} b_k$ are. In general, $\sum_{(t,k)\in\mathbb Z^2} a_tb_k=S$ if for any $\epsilon>0$ there is a finite subset $A\subset \mathbb Z^2$ such that $\sum_{(t,k)\in B}a_tb_k  S<\epsilon$ whenever $B$ is finite and $B\supset A$. Now if both given series converge absolutely, then the contribution from outside of a large square is small, and it follows that $S$ is the product of two sums. 

