nmax=5;
coeffs = Array[F, nmax, 0]; 
F[0] = (z^2 + z (1 + th))/(2 ((Sqrt[z + 1] + th)^2)); Do[
F[k] = Simplify[D[F[k - 1], z]], {k, 1, nmax}]; Do[
F[k] = F[k]/k!, {k, 0, nmax}]; 
Total@Factor[ReplaceRepeated[coeffs, {z -> 0}]]

(* th/(8 (1 + th)^3) + 1/(2 (1 + th)) - (th (3 + th))/(16 (1 + th)^4) *)

(Series[(1 + z + z th)/(2 z ((Sqrt[1 + z] + Sqrt[z] th)^2)), {z, 
       Infinity, 4}] // Simplify) // Normal
(*-((th (3 + th))/(16 (1 + th)^4 z^4)) + th/(8 (1 + th)^3 z^3) + 1/((2 + 2 th) z) *)

Thanks, works for me now.

Your code dosen't work for me ?

Maybe you post a question on mathematica.stackexchange Maybe they can help you there because I don't know how to help you.

Maple result converted to Mathematica code with only 20 terms. Converting is extremely CPU and RAM intensive.

See attached file.

I send you tomorrow series with 30 terms.

Regards M.I.

EDITED: 2020-01-16

I added file: Series only 30 terms.nb .You must change file extension to *.zip and extract file.

Attachments:

Series only 20 terms.nb

Series only 30 terms.nb

My first guess has to be with "s+g" that Mathematica doesn't know if it is two independent variables or a variable and a constant (noting you have separate J(x) - will s or g be constant or in which?). If J is designed for only one variable and tries to "solve" (rather than use table) with two - it could easily just end up in an infinite loop somewhere trying to perform it and waiting for a result it will not get.

Is there a proof the series, using J0(s,g) as you suggest terminates? The book and page citing it?

Fun1[s_] := (1 - BesselK[0, Sqrt[x]]/BesselK[1, Sqrt[x]])/(s*Sqrt[x]);
In[19]:= Series[Fun1[s], {s, \[Infinity], 60}] // Timing

Out[19]= {0.001181, SeriesData[d`s, 
DirectedInfinity[1], {
  d`v^Rational[-1, 2] (1 - BesselK[0, d`v^Rational[1, 2]]/BesselK[
    1, d`v^Rational[1, 2]])}, 1, 61, 1]}

(Is there some reason you cannot, here, say %/.x->(g+s)) ?)

I could ask questions like whether (s*Sqrt[g+s]) ends up being a solution to another besselK but have no time - but I understand such inputs to J are common after converting an ODE to a bessel form ODE by substitution.

I wonder at times why Mathematica doesn't seem to use a larger "table of equations" solutions to produce what (doesn't need to be solved because it's in a table). on the other hand i know some sol'n by mm are by table - but the tables seem to be "brief" when working in certain areas (where try to constrain variables doesn't seem to find more applicable sol'n).