## Mertens’ Theorem

Let and be real or complex sequences.

If the series converges to and converges to , and at least one of them converges absolutely, then their Cauchy product converges to .

An immediate corollary of Mertens’ Theorem is that if a power series has radius of convergence , and another power series has radius of convergence , then their Cauchy product converges to and has radius of convergence at least the minimum of .

Note that a power series converges absolutely within its radius of convergence so Mertens’ Theorem applies.