  Two different sufficient conditions are given for the convergence of the
Magnus expansion arising in the study of the linear differential equation $Y' =
A(t) Y$. The first one provides a bound on the convergence domain based on the
norm of the operator $A(t)$. The second condition links the convergence of the
expansion with the structure of the spectrum of $Y(t)$, thus yielding a more
precise characterization. Several examples are proposed to illustrate the main
issues involved and the information on the convergence domain provided by both
conditions.
