The convergence of bound-state calculations performed via the oscillator basis expansions by means of locating -matrix poles for bound states within the HORSE and SS-HORSE approaches is examined. The convergence in question is studied both in the case of a sharp truncation of the potential matrix in the harmonic-oscillator space and in the case of smoothed matrix elements of the potential. As a result, a new method of extrapolation to the case of the infinite-dimensional model space is proposed. This method makes it possible to predict, on the basis of variational calculations, binding energies and asymptotic normalization coefficients for bound states to a high accuracy and to estimate the uncertainties of these predictions.