Series can not expand the following root object about infinity. Is there another way to expand it?

Question

I have the following equation which I want to solve it for r:

TGB5 = (-((2 q^2)/r^5) + 2/r + (16 P \[Pi] r)/3)/(4 \[Pi]);

rlarge5 = Last[r /. Solve[TGB5 == T, r, Reals]] // Normal

The answer is a root object:

Root[-3 q^2 + 3 #1^4 - 6 [Pi] T #1^5 + 8 P [Pi] #1^6 &, 4]

Now, this function of r should be expanded about T=Infinity. But Mathematica gives the error:

rser5 = Series[rlarge5, {T, Infinity, 4}] // Normal

Series::nmer: Root[-3 q^2+3 #1^4-6 [Pi] T #1^5+8 P [Pi] #1^6&,4] is not a meromorphic function of T at [Infinity].

Is there anyway to expand this root object?

Answer 1

$Version

(* "12.1.0 for Mac OS X x86 (64-bit) (March 14, 2020)" *)

Clear["Global`*"]

TGB5 = (-((2 q^2)/r^5) + 2/r + (16 P π r)/3)/(4 π);

rlarge5 = Last[r /. Solve[TGB5 == T, r, Reals]] // Normal

(* Root[-3 q^2 + 3 #1^4 - 6 π T #1^5 + 8 P π #1^6 &, 4] *)

Since all variables are real and Series takes Assumptions,

rser5 = Assuming[Element[{P, q, T}, Reals],
  Series[rlarge5, {T, Infinity, 4}] // Normal // Simplify]

(* Root::sbr: Because of branch cuts, the series may represent a different 
 root of -3 q^2 T+3 T #1^4-6 π #1^5+8 P π T #1^6& for some values of {P,q,T}.

(8 (-1)^(2/5) 2^(1/5) π^(3/5)
     q^4 (-252 P + 2240 P^3 π^2 q^2 - 225 π T^2) - 
   3840 (-1)^(1/5) 2^(3/5) P^2 π^2 q^4 T^(2/5) Abs[q]^(6/5) - 
   12800 (-1)^(4/5) 2^(2/5) P^2 π^(14/5) q^4 T^(6/5) Abs[q]^(8/5) + 
   2400 P π^(7/5) q^2 T^(4/5) Abs[q]^(12/5) + 
   7200 (-1)^(3/5) 2^(4/5) P π^(11/5) q^2 T^(8/5) Abs[q]^(14/5) - 
   3 (-1)^(1/5) 2^(3/5) T^(
    2/5) (-21 + 3200 P^2 π^2 q^2 + 4000 P π^3 q^2 T^2) Abs[q]^(
    16/5) + 360 2^(2/5) (-π)^(4/5) T^(6/5) Abs[q]^(18/5) + 
   1800 π^(7/5) T^(4/5) (2 P + 5 π T^2) Abs[q]^(22/5) + 
   45000 (-1)^(3/5) 2^(4/5) π^(16/5) T^(18/5) Abs[q]^(
    24/5))/(90000 π^(17/5) T^(19/5) Abs[q]^(22/5)) *)

Version 12.1 finds a series expression; however, this has not considered the conditions for the selected root and issues a warning. Using the ConditionalExpression,

rlarge5r = Last[r /. Solve[TGB5 == T, r, Reals]]

rser5r = Series[rlarge5r, {T, Infinity, 4}] // Normal // Simplify

(* (40 2^(2/5) q^2 (9 (-π)^(4/5) - 320 (-1)^(4/5) P^2 π^(14/5) q^2) T^(
    6/5) + 7200 (-1)^(3/5) 2^(4/5) P π^(11/5) q^2 T^(8/5) Abs[q]^(6/5) - 
   3 (-1)^(1/5) 2^(3/5) T^(
    2/5) (-21 + 4480 P^2 π^2 q^2 + 4000 P π^3 q^2 T^2) Abs[q]^(8/5) + 
   8 (-1)^(2/5) 2^(1/5) π^(
    3/5) (-252 P + 2240 P^3 π^2 q^2 - 225 π T^2) Abs[q]^(12/5) + 
   3000 π^(7/5) T^(4/5) (2 P + 3 π T^2) Abs[q]^(14/5) + 
   45000 (-1)^(3/5) 2^(4/5) π^(16/5) T^(18/5) Abs[q]^(
    16/5))/(90000 π^(17/5) T^(19/5) Abs[q]^(14/5)) *)

This is a slightly simpler form and avoids the warning.

Answer 2

In Mathematica 12.1, the following code gives the right answer:

r /. Last[AsymptoticSolve[TGB5 == T, r, {T, Infinity, 8}]] // Expand

which is:

 -((4 P^2)/(9 \[Pi]^3 T^5)) + (128 P^4 q^2)/(81 \[Pi] T^5) - P/(
 3 \[Pi]^2 T^3) - 1/(2 \[Pi] T) + (3 T)/(4 P)

I do not know why the answer from AsymptoticSolve command is too far from what the Series command computes (Please see Hanlon answer.). Does any one know?