Real series/Cauchy-product/Definition

Cauchy product

For two series ${}\sum _{i=0}^{\infty }a_{i}$ and ${}\sum _{j=0}^{\infty }b_{j}$ of real numbers, the series

$\sum _{k=0}^{\infty }c_{k}{\text{ with }}c_{k}:=\sum _{i=0}^{k}a_{i}b_{k-i}$

is called the Cauchy-product of the series.