The convergence behaviour of expansions in the classical collision theory

In the classical collision theory the scattering angle? depends on the impact parameterb and on the kinetic energyE _r of the relative motion. This angle?(b, E _r ) is expanded for two limiting cases: 1. Expansion in powers of the potentialV(r)/E _r (momentum approximation). 2. Expansion in powers of the impact parameterb (central collision approximation). The radius of convergence of the series depends onb andE _r . It will be given for the following potentialsV(r):

$$Aleft( {frac{a}{r}} right)^mu ;Ae^{ - frac{r}{a}} ;Afrac{a}{r}e^{ - frac{r}{a}} ;Aleft( {frac{a}{r}} right)^2 e^{ - left( {frac{r}{a}} right)^2 } .$$