Convergence of Neumann series

Let \(T\) be an operator. A Neumann series is of the following form

\[\sum_{i = 0}^{\infty}T^{i}.\]

Theorem: Let \(T\) be a bounded linear operator on a Banach space \(X\). If the Neumann series converges in the operator norm, then \(I-T\) is invertible and its inverse is given by

\[(I - T)^{-1} = \sum_{i = 0}^{\infty}T^{i},\]

where \(I\) is the identity operator.

Theorem: