I often hear that divergent series as useful to physics. I read that they appear as asymptotic series in theories like Quantum Field Theory and perturbation theory. In several papers I saw that physicists often like to prove Borel-summability of functions. For example, in Sokal's famous paper generalizing the Watson-Nevanlinna theorem he says that
Using Nevanlinna's theorem, one can simplify the proof of Borel summability of the Schwinger functions in the $\phi_2^4$ quantum field theory"
What good does that do? I do not ask in this specific case, but when you prove the Borel summability of a function, what can you do with it in the the context of physics? And are there any other contexts where divergent series appear or are they essentially all the same?
I'm quite familiar with (the mathematical aspect of) asymptotic series and divergent series in general, but when I'm asked how useful they are, I can't give a satisfactory explanation. I talk about the "Casimir effect" as kind of an argument of authority, not really understanding what's behind.
