  We study the Bose-Einstein condensation phase transition in a weakly
interacting gas through a perturbative analysis of finite systems. In both the
grand canonical and the canonical ensembles, perturbation theory suffers from
infrared divergences and cannot directly determine the transition temperature
in the thermodynamic limit. However, in conjunction with finite size scaling,
perturbation theory provides a powerful calculation tool. We implement it here
to estimate a shift in the transition temperature in the canonical ensemble
consistent with grand canonical calculations.
