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The energy eigenvalues of the quartic potential in a box are obtained exactly and perturbatively using the basis of eigenstates of the free particle in a box. The exact eigenvalues are obtained from the diagonalization of the Hamiltonian in that basis. The perturbative solution is constructed from the Rayleigh-Schroedinger expansion using the quartic potential as the perturbation. The perturbative series is shown to be convergent for small boxes, and an upper bound for the radius of convergence is estimated. Pade-approximant solution, valid for boxes of any size, are also constructed. The numerical comparison of the perturbative and Pade-approximant solutions with the exact ones confirms the validity of the former, especially the convergence and the radius of convergence of the perturbative series. (Atomindex citation 15:003747)