[en] Realistic extended shell-model calculations are used to construct exact effective Hamiltonians, the perturbation series for the effective Hamiltonian to any order, and the [N+1, N] Pade approximants to the series. It is found that the Pade approximants give reliable results even when the series diverge, but that for both convergent and divergent series reasonably accurate results can be obtained only in fifth, or even seventh order. In addition, the poles of the low-order Pade approximants are not always reliable indicators of singularities of the perturbation series. The perturbation series and Pade approximants for the Q-box (energy-dependent effective Hamiltonian) are no more accurate in low orders than those for the usual effective Hamiltonian. Explicit formulas for the matrix Pade approximants are given in an appendix. (Auth.)