解說：

maple code:

>fseries:= proc(f,rng,n)
  local a,b,T,z,sum,k;
  a:=lhs(rhs(rng));
  b:=rhs(rhs(rng));
  T:=b-a; z:=(2*Pi)/(T)*t;
  sum:=(int(f,rng))/T;
  for k from 1 to n do
  sum:=sum+2/T*int(f*cos(k*z),rng)*cos(k*z)+2/T*int(f*sin(k*z),rng)*sin(k*z);
  od;
  sum;
  end:

>f:=t->piecewise(-1<=t and t<0,1,0<=t and t<1,0)

>fs := fseries(f(t), t = -1 .. 1, 30)

> with(plots):
> g1 := plot([fs], t = -1 .. 1, color = red);
   g2 := plot(f(t), t = -1 .. 1, color = blue);
   display(g1, g2);

n=1時

n=3時

n=10時

n=30時