The linear δ expansion is applied to the 3-dimensional O(N ) scalar field theory at its critical point in a way that is compatible with the large-N limit. For a range of the arbitrary mass parameter, the linear δ expansion for φ 2 converges, with errors decreasing like a power of the order n in δ. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with n.

The leading correction from interactions to the transition temperature T c for Bose-Einstein condensation can be obtained from a nonperturbative calculation in the critical O(N ) scalar field theory in 3 dimensions with N = 2. We show that the linear δ expansion can be applied to this problem in such a way that in the large-N limit it converges to the exact analytic result. If the principal of minimal sensitivity is used to optimize the convergence rate, the errors seem to decrease exponentially with the order in the δ expansion. For N = 2, we calculate the shift in T c to fourth order in δ. The results are consistent with slow convergence to the results of recent lattice Monte Carlo calculations.