Neumann series

A Neumann series is a mathematical series of the form

${\displaystyle \sum _{k=0}^{\infty }T^{k}}$

where ${\displaystyle T}$ is an operator and ${\displaystyle T^{k}:={}T^{k-1}\circ {T}}$ its ${\displaystyle k}$ times repeated application. This generalizes the geometric series.

The series is named after the mathematician Carl Neumann, who used it in 1877 in the context of potential theory. The Neumann series is used in functional analysis. It forms the basis of the Liouville-Neumann series, which is used to solve Fredholm integral equations. It is also important when studying the spectrum of bounded operators.