Yes, but you very quickly run out of steam. Moreover, we know the perturbative series does not converge for many theories. For QED, for example, the radius of

Yes, but you very quickly run out of steam. Moreover, we know the perturbative series does not converge for many theories. For QED, for example, the radius of convergence must be 0, by an argument due to Dyson. Roughly: 1. The perturbative series is an analytic function of the fine structure constant α, which is proportional to the electron charge squared. 2. Like charges repel and therefore α>0. If like charges attracted the vacuum would be unstable against the continuous creation of electron-positron pairs from the vacuum and the electrons going to one part of the universe and the positrons to the other. 3. Because of this instability the series converges for no α<0, and since the perturbative expansion is analytic in α cannot converge for any α>0 either.