restart:

with(plots):

f := piecewise(x<0,-1, x>=0,1);

plot(f,x=-1..1);

plot(subs(n=0,LegendreP(n,x)),x=-1..1);

N:=20:

for n from 0 to N do

c[n] := int(f*LegendreP(n,x),x=-1..1)/int(LegendreP(n,x)^2,x=-1..1):

od:

fapprox:=add(c[n]*LegendreP(n,x),n=0..N):

plot([f,fapprox],x=-1..1);

Rel_Err := evalf(sqrt(int((f-fapprox)^2,x=-1..1))/sqrt(int(f^2,x=-1..1)));

Energy:=evalf(int(f^2,x=-1..1));

Energy_Legendre:=evalf(add(c[n]^2*int(LegendreP(n,x)^2,x=-1..1),n=0..N));