(* Built-In Function for Zeros of Bessel Function *)
N[BesselJZero[0,1]]

(* Expression for An's*)
A[n_]=(2/(BesselJZero[0,n] BesselJ[1,BesselJZero[0,n]]))

N[A[1]]

ListPlot[{Table[BesselJZero[0,n],{n,1,10}],Table[BesselJZero[1,n],{n,1,10}]},PlotRange->{0,33}]

ListPlot[Table[ A[n],{n,1,100}],PlotRange-> All]

Atable=Table[N[A[n]],{n,1,100}];

Ztable=Table[N[BesselJZero[0,n]],{n,1,100}];

Plot[Sum[ Atable[[n]] BesselJ[0,Ztable[[n]] ρ],{n,1,1}],{ρ,0,1},PlotRange->All]

Plot[Sum[ Atable[[n]] BesselJ[0,Ztable[[n]] ρ],{n,1,10}],{ρ,0,1},PlotRange->{0,1.20}]

Plot[Sum[ Atable[[n]] BesselJ[0,Ztable[[n]] ρ],{n,1,50}],{ρ,0,1},PlotRange->{0,1.20}]