C. Eminhizer and the authors1have recently developed iterative methods for solving the dynamics of extremely nonlinear anharmonic oscillators. These methods are re‐examined here in a formalism proposed by Carleman2for the embedding of nonlinear dynamical equations in an infinite set of linear equations.The methods are based on recursion formulae for the Fourier coefficients of Fourier series solutions of the dynamical equations and for coefficients of expansions of Fourier frequencies.The series solutions avoid ’’Secular Terms’’ and ’’Small Denominators’’ and converge rapidly in large regimes of highly nonlinear behaviour.