The quasiclassical approximation to the scattering amplitude is expressed, for the first time, as a ratio of absolutely convergent series. We exploit the Fredholm theory of integral equations and the result is shown to be a generalization and simplification of techniques used in resumming the Gutzwiller trace formula. A numerical example is given which displays rapid convergence of the Fredholm series for the scattering amplitude. The method is not restricted to hard chaos or to cases where the curvature expansion is good, but it naturally exploits these simplifications.