The authors consider the anharmonic oscillator Hamiltonian and solve the corresponding Schrodinger equation by means of the recently proposed fixed-point (FP) perturbation theory. The FP formalism resembles an introduction of the creation and annihilation operators, uses a vectorial generalisation of continued fractions, introduces the inverse model-space dimension as a natural and controllably small expansion parameter and is shown to define a 'smooth' (exact, effective) truncation of the Hamiltonian matrix. It also reproduces the recent n>>1 asymptotic formulae for the wavefunctions.