In QFT is said that the renormalized Dyson series is only asymptotic.Bat to be able to say it is necessary to know to what function the Dyson series is asymptotic.
For exemple suppose that some transition amplitude A(g) is given perturbatively by a series of powes of g, in order to prove that this series is asymptotic to A(g) I need to know the value of A(g) not perturbativelly and this is not possible since the only way A(g) is given is the Dyson series