The usual QCD folklore tells us that perturbative expansions are (at best) asymptotic. Recently a colleague of mine told me that the expansion of the beta function is thought to be convergent because , and I quote, "it only sees the UV".
This would-be argument does not convince me at all but I wonder if he really is full of shit or for some reasons that I don't yet comprehend he is right. So, is he right? and if so, why?