If two series $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$ both converge absolutely with sums $A$ and $B$ respectively, the Cauchy product $\sum_{n=1}^{\infty} c_n$ where $c_n = \sum_{k=1}^{n} a_k b_{n+1-k}$ satisfies: