# Matrix Convergence Series

Let $A$ $\in$ $\mathbb{R} ^{n×n}$ and consider the series:

$$S = \sum_{k=0}^{\infty} A^{k}$$
Prove that the series converges iff all the eigenvalues of $A$ are strictly smaller than 1. Further, if the series converges, show that
$S$ is invertible with its inverse being $I − A$.

#### Solutions Collecting From Web of "Matrix Convergence Series"

$$S=I+A+A^2+A^3+A^4+…+A^n\\ \to (I-A)S=\\I(I+A+A^2+A^3+A^4+…+A^n)-A(I+A+A^2+A^3+A^4+…+A^n)=\\I+A+A^2+A^3+A^4+…+A^n-(A+A^2+A^3+A^4+…+A^n+A^{n+1})=\\I-A^{n+1}$$
so now $n \to \infty ,|\lambda_i|<1 \to$ we have $A^{n+1} \to 0$
$$(I-A)S=I-A^{n+1} \to I\\ (I-A)S=I \to S^{-1}=(I-A)\\$$