Let \(\sum_{n=0}^\infty a_n\) and \(\sum_{n=0}^\infty b_n\) be two absolutely convergent complex series with \(A:=\sum_{n=0}^\infty a_n\) and \(B:=\sum_{n=0}^\infty b_n\). We define a complex sequence \((c_n)_{n\in\mathbb N}\) with \[c_n:=\sum_{k=0}^n a_{n-k}b_k.\] Then the series \(\sum_{n=0}^\infty c_n\) is also an absolutely convergent complex series with \(C:=\sum_{n=0}^\infty c_n\). It is called the Cauchy product of the series \(A\) and \(B\): \[C=\sum_{n=0}^\infty c_n=\left(\sum_{n=0}^\infty a_n\right)\cdot\left(\sum_{n=0}^\infty b_n\right)=A\cdot B.\]
Proofs: 1
Proofs: 1