This paper deals with the convergence of series expansions of trajectories for semi-linear infinite dimensional systems, which are analytic in state and affine in input. A special case of such expansions corresponds to Volterra series which are extensively used for the analysis, the simulation and the control of weakly nonlinear finite dimensional systems. The main results of this paper give computable bounds for both the convergence radius and the truncation error of the series. These results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality. They are available for distributed and boundary control systems. As an illustration, these results are applied to an epidemic population dynamic model. In this example, it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues. The relevance of the method is illustrated by simulations.