How to multiply this series:
$$(\sum\_{t=-\infty}^{\infty}a\_{t})(\sum\_{k=-\infty}^{\infty}b\_{k})$$



cauchy product for general case
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.


