  Recently, several studies of non-Hermitian Hamiltonians having $\mathcal{PT}$
symmetry have been conducted. Most striking about these complex Hamiltonians is
how closely their properties resemble those of conventional Hermitian
Hamiltonians. This paper presents further evidence of the similarity of these
Hamiltonians to Hermitian Hamiltonians by examining the summation of the
divergent weak-coupling perturbation series for the ground-state energy of the
$\mathcal{PT}$-symmetric Hamiltonian $H=p^2+{1/4}x^2+i\lambda x^3$ recently
studied by Bender and Dunne. For this purpose the first 193 (nonzero)
coefficients of the Rayleigh-Schr\"odinger perturbation series in powers of
$\lambda^2$ for the ground-state energy were calculated. Pad\'e-summation and
Pad\'e-prediction techniques recently described by Weniger are applied to this
perturbation series. The qualitative features of the results obtained in this
way are indistinguishable from those obtained in the case of the perturbation
series for the quartic anharmonic oscillator, which is known to be a Stieltjes
series.
