Hi, I have again a problem with using Series in connection with Bessel functions. I need an expansion

F[s_]:=(BesselK[1,Sqrt[s]]/(Sqrt[s]*BesselK[0,Sqrt[s]]))/(1+th*Sqrt[d]*
BesselK[1,Sqrt[s]]*BesselK[0,Sqrt[d*s]]/BesselK[0,Sqrt[s]]/BesselK[1,Sqrt[d*s]]);

ser0=Series[F[s],{s,∞,1}]

Unfortunately the above command produces a clearly incompletely performed expansion, since the successive expansion coefficients are functions of s. I have reasons to expect that this is not correct, and that the correct complete expansion contains terms in the form (1/Sqrt[s])^n with coefficients independent of s. In fact, if we perform

Series[SeriesCoefficient[ser0,1/2],{s,∞,1}]

we can see that the first term can still be expanded. Is there any way to obtain a complete expansion, ideally up to 40 terms? Why Series exhibits such a weird behaviour?

Lesław