\begin{definition}[Definition:Cauchy Product]
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:
:$\ds \forall n \in \N: c_n = \sum_{k \mathop = 0}^n a_k b_{n - k} = \sum_{k \mathop = 0}^n a_{n - k} b_k$
{{NamedforDef|Augustin Louis Cauchy|cat=Cauchy}}
\end{definition}