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In this note we show that the standard \mbox{Rayleigh-Schr\"odinger} (RS)perturbation method gives the same result as the hypervirial pertubative method(HPM), for an approximate analytic expression for the energy eigenvalues of thebounded quartic oscillator. This connection between the HPM and the RS methodwent unnoticed for a long time, apparently because it was not obvious that theresulting polygamma sums to be evaluated in the RS method could, in fact, beexpressed in closed form.

An alternative derivation of the Fern\'andez-Castro analytic approximate expression for the eigenvalues of the bounded quartic oscillator