How to resolve problem during summation of functions?

Question

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. "

Answer 1

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]) *)