  The nonperturbative linear delta expansion (LDE) method is applied to the
critical O(N) phi^4 three-dimensional field theory which has been widely used
to study the critical temperature of condensation of dilute weakly interacting
homogeneous Bose gases. We study the higher order convergence of the LDE as it
is usually applied to this problem. We show how to improve both, the large-N
and finite N=2, LDE results with an efficient resummation technique which
accelerates convergence. In the large N limit, it reproduces the known exact
result within numerical integration accuracy. In the finite N=2 case, our
improved results support the recent numerical Monte Carlo estimates for the
critical transition temperature of Bose-Einstein condensation.
