Non-Convergence In Creating Legendre Series

Question

I'm trying to use Mathematica to create a Legendre-Fourier series using this Wikipedia article. Here is my code:

N1=1;
degree=10;

Fun=(-(N1^2 x) + (2 N1^2 x ArcTan[10000 N1^2 x])/Pi)/2;
Coefs=ConstantArray[0,degree];

For[i=0,i<degree,i++,
Legendre=LegendreP[i,x];
f[x_]=Integrate[Fun*Legendre,x];
Coefs[[i+1]]=N[(2*i+1)/2 * f[1]-f[-1]] ;]

LegendreSeries[x_]=Sum[Coefs[[i+1]]*LegendreP[i,x],{i,0,degree-1}];

Plot[{LegendreSeries[x],(-(N1^2 x) + (2 N1^2 x ArcTan[10000 N1^2 x])/Pi)/2},{x,-1,1}]

However, when I went to plot it side-by-side with the original function, even at degree=10, the series fails to converge.

I feel like I did something wrong, but I can't tell what. Could someone help me out?

Answer 1

N1 = 1;
degree = 20;

Fun = 1/2 (-(N1^2 x) + (2 N1^2 x ArcTan[10000 N1^2 x])/π)

al = Table[Integrate[Fun*LegendreP[l, x], {x, -1, 1}]/Integrate[LegendreP[l, x]^2, {x, -1, 1}], {l, 0, degree}];

fn[x_] := N[Sum[al[[l + 1]]*LegendreP[l, x], {l, 0, degree}]]

Plot[{fn[x], 1/2 (-(N1^2 x) + (2 N1^2 x ArcTan[10000 N1^2 x])/\[Pi])}, {x, -1, 1}]