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"