In my Introduction to QFT lecture, we quantized a Klein-Gordon Field and as a toy model we looked at $\phi^3$ theory. For this toy model we expanded the $S = U(-\infty, \infty)$ operator in a series (using Wick's theorem) and then where able to compute observables like scattering cross-sections and decay widths.
I have read a little about the non-existence of the operator $S$ and some other mathematical problems concerning QFT. And for now I tend to accept most of these problems and learn more about them later. But there is one point that really bothers me. I have read that the series expansion of the $S$-matrix is actually "asymptotic" hence the radius of convergence is $0$. The argument what usually follows is that up to some finite order the series still approximates the actual solution$^1$. Okay lets assume that this is actually the case. What is the future of the Standard Model of Particle Physics when all "trustworthy" terms are known? Or even worse: when do we know that the terms are not trustworthy anymore? What if the theory differs from a really precise experiment in the future -- can we distinguish new physics from effects that come from the divergence of the expansion?
$^1$ Is there actually a proof of this statement?