An investigation is made of the convergence of the field-theoretic perturbation expansion of the elements of theS-matrix. For a theory characterized by a coupling linear in the boson field, it is established that if a cut-off is introduced in momentum space, then the transition amplitude for any given process over a finite-time interval, is bounded term by term by an exponential series. The extension of the method to interactions which are not linear, is, for several cases, also included. The implications of the result for theories without a cut-off is discussed from a qualitative point of view.