There are considered a Banach space \(X\), a family of linear operators \((A(t), D(A(t)))\) and the Cauchy problem \[ \begin{cases}{\frac{d}{dt}}u(t)=A(t)u(t), \quad t\geq s\in {\mathbb R},\\ u(s)=x\in X. \end{cases} \] Among the numerical methods for the approximation of the solution, the Magnus integrators play an important role. The basic idea of this method is to express the solution \(u(t)\) in the form \(u(t)=\text{exp}\; \Omega (t)\cdot x\) with \(s=0\). Here, \(\Omega (t)\) is an infinite sum yielded by a Picard iteration. The authors investigate the convergence of a certain Magnus method, where they take the first term of the Magnus series expansion. They prove that the method converges in the operator norm and has a uniform convergence order.