You should take a look at Francisco Fernandez "Introduction to Perturbation Theory in Quantum Mechanics", in chapter 6 he notes that almost all perturbation series are divergent, including fan-favorites Stark and Zeeman effect, so that having a convergent series is an exception. This has nothing to do with the infinite degrees of freedom of QFT, is just a fact of QM.
As an addendum, let me just note that although almost all perturbation series are indeed divergent the problem only arises (or arised) in QFT. That's because in finite degrees of freedom QM we can define rigorously the hamiltonian, for example the Zeeman effect one, we just can't compute exactly the eigenvalues.
In the early days of QFT people only knew how to define free fields, there was no satisfactory definition of a interacting QFT. For some time there was some hope of defining QFT to BE the results of perturbation series. That's why Dyson's argument was significant, it shows that one cannot define interacting QFT via perturbation expansion.
(The last couple of paragraphs we're my extrapolation of why you we're curious about convergence in QM, and so I thought appropriate to include. I'll be happy to delete them if people find it not pertaining to the questions as such)
This post imported from StackExchange Physics at 2017-10-16 12:24 (UTC), posted by SE-user cesaruliana