Cauchy Product

Let $a_n$ and $b_n$ be two sequences. The Cauchy product of these series is the new series $c_n$

$$\left(\sum _{n=0}^{\infty }{a}_n\right)\left(\sum _{n=0}^{\infty }{b}_n\right)=\sum _{n=0}^{\infty }{c}_n.$$

where

$$c_n=\sum _{k=0}^n{a}_k{b}_{n-k}.$$

That is, each coefficient $c_n$ is obtained by summing all products $a_k{b}_{n-k}$ whose indices add up to $n$. If both original series converge absolutely, then the Cauchy product also converges absolutely.