Time-dependent perturbation theory as a tool to compute approximate solutions of the Schrodinger equation does not preserve unitarity. Here we present, in a simple way, how the "Magnus expansion" (also known as "exponential perturbation theory") provides such unitary approximate solutions. The purpose is to illustrate the importance and consequences of such a property. We suggest that the Magnus expansion may be introduced to students in advanced courses of quantum mechanics. (Contains 1 figure.)