The Born series applied to the Lippmann-Schwinger equation is a straightforward method for solving optical scattering problems, which however diverges except for very weak scatterers. Replacing the Born series by Padé approximants is a solution of this problem. However, in some cases it is rather difficult to obtain an accurate Padé approximant. In this paper we aim to understand the cause by studying the scattering by a cylinder. We find that there is a strong connection between eigenvalues of the Lippmann-Schwinger operator that are close to the real axis, the occurrence of a near-resonance, and the problematic behavior of the Padé approximant. The determination of these eigenvalues provides a general method to obtain, for any given geometry, materials for which near-resonances occur.@en