If you replace Legendre P_n by its asymptotic expansion, convert the cosines into terms of e^{in\theta}, and split the series into a series of each term, it gets the desired result:
pn = Series[ (* get asymptotic expansion of LegendreP[] *)
LegendreP[n, Cos[\[Theta]]],
{n, Infinity, 0},
Assumptions -> (0 < \[Theta] < \[Pi] && n \[Element] Integers &&
n > 0)];
Assuming[0 < \[Theta] < \[Pi] && n \[Element] Integers && n > 0,
Normal[pn] // Simplify //
ReplaceAll[Sin[\[Theta]] -> s] // (* get around TrigToExp *)
TrigToExp //(* SumConvergence knows more about Exp[I n \[Theta]] *)
Expand // (* expand into sum of terms *)
Apply@List // (* split terms *)
ReplaceAll[s -> Sin[\[Theta]]] // (* put back Sin[] *)
(*Echo//*) (* place Echo[] to show any intermediate result *)
SumConvergence[(* test convergence of series of each term *)
t^n #, n,
Assumptions -> (0 < \[Theta] < \[Pi] && 0 < s <= 1 &&
n \[Element] Integers && n > 0)] & //
Apply@And // (* require all series to converge *)
Simplify] (* simplify And[...] *)
(* t == -1 || t == 1 || Abs[t] < 1 *)
Thanks a lot, Michael! It worked. However, when it comes to the equation for Subscript[Ar[Theta], 3] in the previously attached notebook—which involves the multiplication of a Bessel function and a Legendre function—Mathematica gets stuck and becomes unresponsive. I am attaching the updated notepook with this reply. Could you also help me with this?
[Note: I don’t like using Subscript[] for variables. For instance, the expressions Subscript[r, 0] and r both depend on r, just as f[r, 0] and r both depend on r; and I suspect they are not supposed to.]
Using some asymptotic expansions from https://dlmf.nist.gov, we get it fails to converge:
(*****)
oldAssumptions = $Assumptions;
Protect[oldAssumptions];
$Assumptions =
0 < \[Theta] < \[Pi] && l \[Element] Integers && l >= 0 && r > 0 &&
r0 > 0 && \[Gamma] > 0;
(* "https://dlmf.nist.gov/10.19.E1" *)
asympBesselJ = BesselJ[n_, z_] :> 1/Sqrt[2 Pi n] ((E z)/(2 n))^-n;
(* "https://dlmf.nist.gov/14.15.E11" *)
asympLegendreP = LegendreP[n_, m_, Cos[t_]] :>
1/n^-m (t/Sin[t])^(1/2) BesselJ[-m, (n + 1/2) t];
res1 = Series[((-1)^l (2^(-(5/2) - l)) Sqrt[\[Pi]]
BesselJ[3/2 + 2 l,
r \[Gamma]] (-1 + 2 l)!! (3 + 4 l)!!*((\[Gamma] r0)^(-1 -
2 l)) )/(Sqrt[r \[Gamma]] Gamma[2 + l])
LegendreP[2*l + 1, 1, Cos[\[Theta]]] /. asympLegendreP /.
asympBesselJ, {l, Infinity, 0},
Assumptions -> (0 < \[Theta] < \[Pi] && l \[Element] Integers &&
l >= 0)] // Normal
SumConvergence[res1, l]
(* False <-- RESULT *)
$Assumptions = oldAssumptions;
(*****)
Maybe there’s an error in the formulas…
I double-checked the equation, and it was correct. The only issue I realized was that the asymptotic forms for Bessel functions from the link you provided are valid for positive real values, which is not the case in my situation.
I was unable to resolve the issue despite trying various ways.