Abstract: I revisit the question of convergence of the KSW expansion brought into question by the paper of Fleming, Mehen and Stuart (1999). In that paper the authors saw poor convergence at second order in the expansion for the spin-triplet channel, and noted that the problems seemed to arise from certain graphs, persisting in the chiral limit. In order to investigate this more closely, I reconsider that class of problematic graphs, equivalent to solving the Schrodinger equation perturbatively in the chiral limit. I show that it is possible to turn the calculation into an iterative algebraic problem, allowing one to easily compute the scattering amplitudes in all partial waves to high order. This allows one to see clearly where the expansion breaks down, and to connect it to observations by Birse who used atomic physics techniques to determine the radius of convergence of the expansion in each channel. This work suggests that one should consider a generalization of the renormalization group to a sort of "running angular momentum" and clarifies mathematically what an improvement of the KSW expansion must achieve -- the explicit removal of a branch cut -- to extend the validiity of the expansion.