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.