Cauchy Product - Convergence and Mertens' Theorem

Convergence and Mertens' Theorem

This section is not about Mertens' theorems concerning distribution of prime numbers.

Let and be real sequences. It was proved by Franz Mertens that if the series converges to B and the series converges absolutely to A then their Cauchy product converges to AB. It is not sufficient for both series to be conditionally convergent. For example, the sequences are conditionally convergent but their Cauchy product does not converge.