This paper presents a series expansion for the evolution of a class of nonlinear systems characterized by constant input vector fields. We present a series expansion that can be computed via explicit recursive expressions, and we derive sufficient conditions for uniform convergence over the finite-and infinite-time horizon. Furthermore, we present a simplified series and convergence analysis for the setting of second-order polynomial vector fields. The treatment only relies on elementary notions on analytic functions, number theory, and operator norms.