What is the discrete analogue of the Liouville–Neumann series? The Liouville-Neumann seres is a manner of writing the solution to a Fredholm integral equation as an infinite series. What is the discrete analogue finite dimensional matrices?

Simply: if you want to solve $(I - \lambda A) x = y$, and $\lambda$ is sufficiently small, then $I - \lambda A$ can be inverted using a geometric series

$$x = \sum_{k=0}^{\infty} \lambda^k A^k y.$$

The Liouville-Neumann series is exactly this but for $A$ an integral operator instead of a matrix.