Neumann series

If

a

belongs to a unital Banach algebra and

∥ a ∥ < 1

, then

(1 - a)^{- 1} = \sum_{n = 0}^{\infty} a^{n} .

Absolute convergence and multiplication of partial sums prove the identity.

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The Neumann series is a mathematical series used to represent the inverse of an operator (or a matrix) under certain conditions related to convergence. Specifically, it is often utilized in functional analysis and linear algebra. The Neumann series is particularly useful when dealing with bounded linear operators in Hilbert or Banach spaces, as well as with matrices.

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