Proposition: Cauchy Product of Absolutely Convergent Complex Series

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

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983