Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent

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 convergent1.

Proofs: 1

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


  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.