A Neumann series is a mathematical series that sums k-times repeated applications of an operator ${\displaystyle T}$. This has the generator form

where ${\displaystyle T^{k}}$ is the k-times repeated application of ${\displaystyle T}$; ${\displaystyle T^{0}}$ is the identity operator ${\displaystyle I}$ and ${\displaystyle T^{k}:={}T^{k-1}\circ {T}}$ for ${\displaystyle k>0}$. This is a special case of the generalization of a geometric series of real or complex numbers to a geometric series of operators. The generalized initial term of the series is the identity operator ${\displaystyle T^{0}=I}$ and the generalized common ratio of the series is the operator ${\displaystyle T.}$

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 is closely connected to the resolvent formalism for studying the spectrum of bounded operators and, applied from the left to a function, it forms the Liouville-Neumann series that formally solves Fredholm integral equations.

Suppose that ${\displaystyle T}$ is a bounded linear operator on the normed vector space ${\displaystyle X}$. If the Neumann series converges in the operator norm, then ${\displaystyle I-T}$ is invertible and its inverse is the series:

where ${\displaystyle I}$ is the identity operator in ${\displaystyle X}$. To see why, consider the partial sums

Then we have

This result on operators is analogous to geometric series in ${\displaystyle \mathbb {R} }$.

One case in which convergence is guaranteed is when ${\displaystyle X}$ is a Banach space and ${\displaystyle |T|<1}$ in the operator norm; another compatible case is that ${\textstyle \sum _{k=0}^{\infty }|T^{k}|}$ converges. However, there are also results which give weaker conditions under which the series converges.