Isn't it a little more subtle than this makes it sound? Because as the order goes to infinity, the perturbation series is not to be expected to approximate the true physical value, but to be infinitely far from it! The perturbation series of realistic QFTs is expected to be divergent (this goes back to Dyson 52.) Now for a general divergent formal power series, even summing up the first few terms makes no particular sense. But since we may assume that the perturbation series is the Taylor expansion of an actual smooth function (namely the non-perturbative theory) it is plausible to expect that it is, even if not convergent, an "asymptotic series". If so, this sort of guarantees that the first few terms (depending on "how small Plnack's constant really is") give a good approximation, but it still means that beyond that the series will diverge arbitrarily far from the desired physical value.