The Cauchy product of two series
and
defined as
with
converges to
where
and
if and only if at least one of
is absolutely convergent.
Proof: Define
For each
Define
Since 
converges to
it is enough to show that
converges to zero. Choose
then let
and 
There is
such that
implies that
and for
thus for
Also, ifconverges to
converges to
converges to
then