[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;
(*****)
