In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section. Cauchy product of two infinite series Let and be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows: where . Cauchy product of two power series Consider the following two power series and with complex coefficients and . The Cauchy product of these two power series is defined by a discrete convolution as follows: where . Convergence and Mertens' theorem Let and be real or complex sequences. It was proved by Franz Mertens that, if the series converges to and converges to , and at least one of them converges absolutely, then their Cauchy product converges to . The theorem is still valid in a Banach algebra (see first line of the following proof). It is not sufficient for both series to be convergent; if both sequences are conditionally convergent, the Cauchy product does not have to converge towards the product of the two series, as the following example shows: Example Consider the two alternating series with which are only conditionally convergent (the divergence of the series of the absolute values follows from the direct comparison test and the divergence of the harmonic series). The terms of their Cauchy product are given by for every integer . Since for every we have the inequalities and , it follows for the square root in the denominator that , hence, because there are summands, for every integer . Therefore, does not converge to zero as , hence the series of the diverges by the term test. Proof of Mertens' theorem For simplicity, we will prove it for complex numbers. However, the proof we are about to give is formally identical for an arbitrary Banach algebra (not even commutativity or associativity is required). Assume without loss of generality that the series converges absolutely. Define the partial sums with Then by rearrangement, hence Fix . Since by absolute convergence, and