  It is well known that quantum-mechanical perturbation theory often give rise
to divergent series that require proper resummation. Here I discuss simple ways
in which these divergences can be avoided in the first place. Using the
elementary technique of relaxed fixed-point iteration, I obtain convergent
expressions for various challenging ground states wavefunctions, including
quartic, sextic and octic anharmonic oscillators, the hydrogenic Zeeman
problem, and the Herbst-Simon Hamiltonian (with finite energy but vanishing
Rayleigh-Schr\"odinger coefficients), all at arbitarily strong coupling. These
results challenge the notion that non-analytic functions of coupling constants
are intrinsically "non-perturbative". A possible application to exact
diagonalization is briefly discussed.
