In the perturbative approach to field theory we expand whatever we are computing on a power expansion in some coupling $$ \sum^nd_n\alpha^n $$ then in principle we can compute all the $d_n$. This series is in general expected not to be convergent, but it is hoped that it at least is an asymptotic expansion of the true thing when $\alpha\to0$.
My question is, why is the perturbative approach only implemented around $\alpha=0$? I mean, we could make expansions around any given $\alpha_0$ and obtain expansions that (would be hoped) to be asymptotic to the real thing when $\alpha\to\alpha_0$? $$ \sum^nb_n(\alpha-\alpha_0)^n $$ why is perturbation theory always implemented around $\alpha=0$?