We investigate the behavior of self-propelled particles in infinite space dimensions by comparing two powerful approaches in many-body dynamics: the Fokker-Planck equation and dynamical mean-field theory. The dynamics of the particles at low densities and infinite persistence time is solved in the steady state with both methods, thereby proving the consistency of the two approaches in a paradigmatic out-of-equilibrium system. We obtain the analytic expression for the pair distribution function and the effective self-propulsion to first-order in the density, confirming the results obtained in a previous paper [T. Arnoulx de Pirey et al., Phys. Rev. Lett. 123, 260602 (2019)] and extending them to the case of a non-monotonous interaction potential. Furthermore, we obtain the transient behavior of active hard spheres when relaxing from the equilibrium to the nonequilibrium steady state. Our results show how collective dynamics is affected by interactions to first-order in the density and point out future directions for further analytical and numerical solutions of this problem.