Let $a_k$ and $b_k$ be two sequences of real or complex numbers for $k \in {\mathbb N}_0$ ( ${\mathbb N}_0$ is the set of natural numbers containing zero). The Cauchy product is defined by: \begin{equation} (a \circ b)(k) = \sum_{l=0}^k a_l b_{k-l}. \end{equation}This is basically the convolution for two sequences. Therefore the product of two series $\sum_{k=0}^{\infty} a_k$ $\sum_{k=0}^{\infty} b_k$ is given by: \begin{equation} \sum_{k=0}^{\infty} c_k = \left(\sum_{k=0}^{\infty} a_k \right)\cdot \left(\sum_{k=0}^{\infty} b_k \right) = \sum_{k=0}^{\infty} \sum_{l=0}^k a_l b_{k-l}. \end{equation}A sufficient condition for the resulting series $\sum_{k=0}^{\infty} c_k$ to be absolutely convergent is that $\sum_{k=0}^{\infty} a_k$ and $\sum_{k=0}^{\infty} b_k$ both converge absolutely .