Cauchy Product

analysis

Definition

Cauchy Product

Due to the vector space property of the space of convergent sequences, convergent series can be added and multiplied by scalars.

For convergent series ${\sum }_n{a}_n$ and ${\sum }_n{b}_n$, the following holds:

$$\begin{array}{rl}\sum _n(a_n+b_n)& =\sum _n{a}_n+\sum _n{b}_n\\ \forall \lambda \in \mathbb{R}:\sum _n\lambda a_n& =\lambda \sum _n{a}_n\end{array}$$

The Cauchy product of two series ${\sum }_{n\ge 0}{a}_n$ and ${\sum }_{n\ge 0}{b}_n$ is the series ${\sum }_{n\ge 0}\left({\sum }_{k=0}^n{a}_k{b}_{n-k}\right)$.