There is a theory of [asymptotic series](https://en.wikipedia.org/wiki/Asymptotic_series). I guess you know that, but the usual perturbation series in quantum field theory do not converge. For example the scattering amplitudes for QED in their series representation given by the Feynman rules are known to be asymptotic series (and (perturbative) QED is not “late 20th century”, but about 60 years old.

I would not say that this is not very common: QFT and string theory always have to deal with such series. You might say that this is what physicists do, but not mathematicians. But in dealing with such series there are hard mathematical problems involved, that are considered by mathematicians. But modern mathematics are of course always “definition based” (well, sometimes even in mathematics the “right definitions” have to be found, but it is not like in theoretical physics where people are often not even seriously looking for the definition)—you _define_ series as formal objects, which you can manipulate although they do not converge.