# Neumann series

If $A$ is a square matrix, $\|A\|<1$, then $I-A$ is nonsingular and $(I-A)^{-1}=I+A+A^{2}+\cdots=\sum_{k=0}^{\infty}A^{k}$. This is the Neumann series.
It provides approximations of $(I-A)^{-1}$ when $A$ has entries of small magnitude. For example, a first-order approximation is $(I-A)^{-1}\approx I+A$.
It is obvious that this is a generalization of the geometric series.

## References

Title Neumann series NeumannSeries 2013-03-22 15:25:49 2013-03-22 15:25:49 georgiosl (7242) georgiosl (7242) 9 georgiosl (7242) Theorem msc 15-00