# 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:

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

## Source of Name

This entry was named for Augustin Louis Cauchy.