Let \(\sum_{n=0}^\infty a_n\) and \(\sum_{n=0}^\infty b_n\) be two convergent series. For a fixed index \(n\), we set
\[c_n:=\sum_{k=0}^n a_{n-k}b_k.\]
Then the series \(\sum_{n=0}^\infty c_n\), also called the Cauchy product of the initial two series, is not necessarily convergent^{1}.
Proofs: 1
A sufficient condition for the Cauchy product to converge is the absolute convergence of the initial two series - see corresponding proposition. In this case, the Cauchy product is even absolutely convergent. ↩