Definition:Cauchy Product

Definition
Let $A := \ds \sum_{n \mathop = 0}^\infty a_n$ and $B := \ds \sum_{n \mathop = 0}^\infty b_n$ be two series.

The Cauchy product $C$ of $A$ and $B$ is defined as:
 * $\ds C := \sum_{n \mathop = 0}^\infty c_n = \sum_{n \mathop = 0}^\infty a_n \cdot \sum_{n \mathop = 0}^\infty b_n$

where:
 * $\ds \forall n \in \N: c_n = \sum_{k \mathop = 0}^n a_k b_{n - k} = \sum_{k \mathop = 0}^n a_{n - k} b_k$