base. The induction step goes as follows: Let the claim be true for an such that , and let be infinite series with complex coefficients, from which all except the th one converge absolutely, and the -th one converges. We first apply the induction hypothesis to the series . We obtain that the series converges, and hence, by the triangle inequality and the sandwich criterion, the series converges, and hence the series converges absolutely. Therefore, by the induction hypothesis, by what Mertens proved, and by renaming of variables, we have: Therefore, the formula also holds for . Relation to convolution of functions A finite sequence can be viewed as an infinite sequence with only finitely many nonzero terms, or in other words as a function with finite support. For any complex-valued functions f, g on with finite support, one can take their convolution: Then is the same thing as the Cauchy product of and . More generally, given a monoid S, one can form the semigroup algebra of S, with the multiplication given by convolution. If one takes, for example, , then the multiplication on is a generalization of the Cauchy product to higher dimension. Notes References . . . . . . . . . . . . . External links . . Augustin-Louis Cauchy Real analysis Complex analysis Sequences and series Articles containing proofs