Summary: The freedom of choice of the zero-order term in the perturbative analysis of harmonic oscillators that are perturbed by a nonlinear perturbation is investigated in detail within the framework of the method of normal forms in the case of the unforced Duffing oscillator. It is demonstrated that the choice leading to minimal normal forms is by far the best, indicating that minimal normal forms may be a way to significantly improve the convergence properties of the perturbation series relative to the traditional expansions.