> plot(Heaviside(t),t=-Pi..Pi);

> int(Heaviside(t),t=-Pi..Pi);
                                     Pi
> int(Heaviside(t)*sin(t),t=-Pi..Pi);
                                      2
> int(Heaviside(t)*sin(2*t),t=-Pi..Pi);
                                      0
> int(Heaviside(t)*sin(3*t),t=-Pi..Pi);
                                      2
                                      -
                                      3
> int(Heaviside(t)*sin(4*t),t=-Pi..Pi);
                                      0
> int(Heaviside(t)*sin(5*t),t=-Pi..Pi);
                                      2
                                      -
                                      5
> int(Heaviside(t)*sin(7*t),t=-Pi..Pi);
                                      2
                                      -
                                      7
> plot(Pi/Pi/2+2*sin(t)/Pi+2/3*sin(3*t)/Pi+2/5*sin(5*t)/Pi+2/7*sin(7*t)/Pi+2/9*sin(9*t)/Pi,t=-Pi..Pi);

> Pi/Pi+2*sin(t)+2/3*sin(3*t)/Pi+2/5*sin(5*t)/Pi+2/7*sin(7*t)/Pi;
                            2 sin(3 t)   2 sin(5 t)   2 sin(7 t)
             1 + 2 sin(t) + ---------- + ---------- + ----------
                               3 Pi         5 Pi         7 Pi   
> sum(2/(2*k+1)*sin((2*k+1)*t)/Pi,k=0..6);
 2 sin(t)   2 sin(3 t)   2 sin(5 t)   2 sin(7 t)   2 sin(9 t)   2 sin(11 t)
 -------- + ---------- + ---------- + ---------- + ---------- + -----------
    Pi         3 Pi         5 Pi         7 Pi         9 Pi         11 Pi   

      2 sin(13 t)
    + -----------
         13 Pi   
> plot(1/2+sum(2/(2*k+1)*sin((2*k+1)*t)/Pi,k=0..100),t=-Pi..Pi);

>