Theorem, Inverse of Matrix is the Sum of Power I noticed that $(1+S)^{-1}=\sum_{n=0}^{\infty}S^n$, where S is a square matrix. Is there any theorem related this identity?

This is only true (i.e., converges), if each eigenvalue of $S$ has absolute value less than $1$. Is has been discussed often on MSE, e.g.,here, or here. If $S$ is nilpotent with $S^n=0$, then
$$
(1+S)^{-1}=I-S+S^2\cdots \pm S^{n-1}.
$$
