Using a truncated Born expansion of the Green tensor, we consider the spontaneous decay rate of an excited two-level atom placed in the vicinity of a finite-length dielectric cylinder. It is shown that the spontaneous decay rate experiences most pronounced modifications in the edge regions. By comparing the results obtained for finite-length cylinders with those obtained for an infinitely extended one, we establish the parameters ranges where the latter is a valid approximation. We discuss how some typical singularities in the Born series, which arise when the integral variables coincide, can be removed.