Geometric Series

Parent: Series

A geometric series is an infinite sum of terms with a fixed ratio between terms:

\[\sum_{}\]

Neumann Series

Neumann series generalize geometric series to linear operators. Specifically, let \(T\) be a linear operator and consider the series:

\[\sum_{k=0}^{\infty} T^k\]

If the Neumann series converges, then the series can be written as

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

Proof: Define \(X := \sum_{k=0}^{\infty} T^k\). Then \(T X = \sum_{k=0}^{\infty} T^{k - 1}\). The difference \(X - TX = (I - T)X = T^0 = I \Rightarrow X = (I - T)^{-1}\).