The issue isn't that an infinite sum of tiny terms don't converge -- the issue is that individual terms of perturbation theory diverge. An example can be found

The issue isn't that an infinite sum of tiny terms don't converge -- the issue is that individual terms of perturbation theory diverge. An example can be found in J. Chem. Phys. 112, 2000, 9736-9748 "Divergence in Moller--Plesset Theory: A Simple Explanation Based on a Two-State Model" DOI 10.1063/1.481611 (Note that this is specifically in reference to Moller--Plesset Perturbation Theory, but the divergence is a general phenomenon) I'm not saying that all perturbation theories diverge. Moller--Plesset perturbation theory doesn't even always diverge. But the divergence behaviour is not in the form of an infinite sum of tiny terms being infinite, but rather the individual terms of the perturbation theory increasing without bound (and oscillating sign). Also note that it is possible for truncations of perturbation theory to diverge with increasing order, but for the infinite sum of all (divergent) PT terms to converge and be finite.