The Character of Convergence of the Cauchy Product

Abstract

Abstract In general, the Cauchy product of an absolutely convergent series and a conditionally convergent one might converge absolutely. In our note, we provide an easy and quite general method for construction of such pairs of series, a method that is not related to the classic Pringsheim’s example. Moreover, we observe that when only pairs of alternating series, both satisfying the assumptions of the alternating series test are considered, if one of them is absolutely convergent then the character of convergence of their Cauchy product is exactly the same as the character of convergence of the second factor. We complete the remarks with a new and surprisingly short proof of the Voss Theorem on Cauchy products.