On the divergence of an asymptotic expansion of a solution to the third Painlevé equation

We find formal asymptotics for solutions of the third Painlevé equation near infinity, in particular, a formal Puiseux series expansion. This series is an asymptotic expansion of a genuine solution and belongs to the Gevrey class of order 1 (after reducing it to a power series). We obtain a family of the equation parameters such that estimates for the coefficients of the Gevrey series under consideration are exact, hence the series diverges.