# How to resolve problem during summation of functions?

I am trying to find out the output of this basic problem but getting an error. If anyone can resolve this will be helpful.

x1 = 1;
w1 = 2;
j = Range[0, 10];
Subscript[a1, j] = Sqrt[Pi*x1/2]*BesselJ[j + 1/2, x1];
Subscript[b1, j] = Sqrt[Pi*w1/2]*BesselJ[j + 1/2, w1];
R1 = \!$$\*UnderoverscriptBox[\(\[Sum]$$, $$j = 1$$, $$10$$]$$\((2*j + 1)$$*Re[
$$\*SubscriptBox[\(a1$$, $$j$$]\)[$$[j]$$] +
$$\*SubscriptBox[\(b1$$, $$j$$]\)[$$[j]$$]]\)\)

Error:
art::partw: "Part 3 of Subscript[a1, 3] does not exist. "


Giving values to Subscript is not a preferred way in Mathematica. Instead, define $$a_1$$ and $$b_1$$ as they were "functions":

Clear[x1, w1, a1, b1, R1];
x1 = 1;
w1 = 2;
a1[j_] := Sqrt[Pi*x1/2]*BesselJ[j + 1/2, x1];
b1[j_] := Sqrt[Pi*w1/2]*BesselJ[j + 1/2, w1];
R1 = \!$$\*UnderoverscriptBox[\(\[Sum]$$, $$j = 1$$, $$10$$]$$\((2*j + 1)$$*
Re[a1[j] + b1[j]]\)\)

(* 3 (Sqrt[\[Pi]/2] BesselJ[3/2, 1] + Sqrt[\[Pi]] BesselJ[3/2, 2]) +
5 (Sqrt[\[Pi]/2] BesselJ[5/2, 1] + Sqrt[\[Pi]] BesselJ[5/2, 2]) +
7 (Sqrt[\[Pi]/2] BesselJ[7/2, 1] + Sqrt[\[Pi]] BesselJ[7/2, 2]) +
9 (Sqrt[\[Pi]/2] BesselJ[9/2, 1] + Sqrt[\[Pi]] BesselJ[9/2, 2]) +
11 (Sqrt[\[Pi]/2] BesselJ[11/2, 1] + Sqrt[\[Pi]] BesselJ[11/2, 2]) +
13 (Sqrt[\[Pi]/2] BesselJ[13/2, 1] + Sqrt[\[Pi]] BesselJ[13/2, 2]) +
15 (Sqrt[\[Pi]/2] BesselJ[15/2, 1] + Sqrt[\[Pi]] BesselJ[15/2, 2]) +
17 (Sqrt[\[Pi]/2] BesselJ[17/2, 1] + Sqrt[\[Pi]] BesselJ[17/2, 2]) +
19 (Sqrt[\[Pi]/2] BesselJ[19/2, 1] + Sqrt[\[Pi]] BesselJ[19/2, 2]) +
21 (Sqrt[\[Pi]/2] BesselJ[21/2, 1] + Sqrt[\[Pi]] BesselJ[21/2, 2]) *)