The matrix exponential of a real matrix $M \in \R ^{n \times n}$ is the matrix \[ I + M + \frac{M^2}{2!} + \frac{M^3}{3!} + \cdots. \] It is a (nontrivial) fact that this (matrix) power series converges for all $M \in \R ^{n \times n}$.
\[ I + M + \frac{M^2}{2!} + \frac{M^3}{3!} + \cdots. \]