Product of Absolutely Convergent Series

Theorem
Let $\map f z = \ds \sum_{n \mathop = 1}^\infty a_n$ and $\map g z = \ds \sum_{n \mathop = 1}^\infty b_n$ be two real or complex series that are absolutely convergent.

Then $\map f z \map g z$ is an absolutely convergent series, and:


 * $\map f z \map g z = \ds \sum_{n \mathop = 1}^\infty c_n$

where:
 * $c_n = \ds \sum_{k \mathop = 1}^n a_k b_{n - k}$