Context. We investigate periodic orbits in galactic potentials by developing analytical methods. Aims. We evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. Methods. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series, a resummation based on a continued fraction may be performed. This method is analogous to the Prendergast method, which searches for approximate rational solutions. Results. It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct accurate solutions both for normal modes and periodic orbits in general position.