Summary: It is often assumed that bound states of quantum mechanical systems are intrinsically non-perturbative in nature and therefore any power series expansion methods should be inapplicable to predict the energies for attractive potentials. However, if the spatial domain of the Schrödinger Hamiltonian for attractive one-dimensional potentials is confined to a finite length L, the usual Rayleigh-Schrödinger perturbation theory can converge rapidly and is perfectly accurate in the weak-binding region where the ground state’s spatial extension is comparable to L. Once the binding strength is so strong that the ground state’s extension is less than L, the power expansion becomes divergent, consistent with the expectation that bound states are non-perturbative. However, we propose a new truncated Borel-like summation technique that can recover the bound state energy from the diverging sum. We also show that perturbation theory becomes divergent in the vicinity of an avoided-level crossing. Here the same numerical summation technique can be applied to reproduce the energies from the diverging perturbative sums.