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 $X - TX = (I - T)X = T^0 = I \Rightarrow X = (I - T)^{-1}$.