Resumming the pressure

The convergence properties of the resummed thermal perturbation series for the thermodynamic pressure are investigated by comparison with the exact results obtained in large-N phi^4 theory and possibilities for improvements are discussed. By going beyond conventional resummed perturbation theory, renormalization has to be carried out nonperturbatively yet consistently. This is exemplified in large-N phi^4_4 and in a special large-N \phi^3_6 model that mimics QED in the limit of large flavour number.