For certain systems, there indeed are convergent perturbation series. In the case of the anharmonic oscillator described by a Hamiltonian,
$$H = \frac12 p^2 + \frac12 m^2x^2 + \frac14 gx^4$$
one may construct a convergent series which is convergent for any $g>0$ and arbitrary harmonic term, valid in both the weak and strong coupling limits.
Similarly, using a procedure involving splitting the Hamiltonian in a particular way and applying perturbation theory, a system of coupled harmonic oscillators admitted a convergent series.
Furthermore, there exists a generalisation of a convergent perturbation series for a $q$-deformed anharmonic oscillator, which is to say a system based on a $q$-deformed Heisenberg algebra, with a Hamiltonian based on operators with modified commutation relations.
In a more general setting, it has been shown that for a class of hyperbolic differential equations, there exists convergent perturbation series for particular families under certain conditions. (As an aside this is used to show the Einstein field equations in a particular scheme yield a divergent perturbation series as opposed to asymptotic.)
This post imported from StackExchange Physics at 2017-10-16 12:24 (UTC), posted by SE-user JamalS