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The energy eigenvalues of harmonic oscillator in circular and spherical boxes are obtained through the Rayleigh-Schroedinger perturbative expansion, taking the free particle in a box as the non-perturbed system. The perturbative series is shown to be convergent for small boxes, and an upper bound for the radius of convergence is established. Pade-approximant solutions are also constructed for boxes of any size. Numerical comparison with the exact eigenvalues - which are obtained by constructing and diagonalising the Hamiltonian in the basis of the eigenfunctions of the free particle in a box - corroborates the accuracy and range of validity of the approximate solutions, particularly the convergence and the radius of convergence of the perturbative series. (Atomindex citation 15:003748)