restart:

with(plots):

Odd and even extensions of functions: rate of convergence

f_odd := (x,N) -> 2/Pi*add((-1)^(n+1)/n*sin(n*Pi*x),n=1..N):

f_even := (x,N)-> 1/2 - 4/Pi^2*add(cos((2*n-1)*Pi*x)/(2*n-1)^2,n=1..N):

plot([f_odd(x,10),f_even(x,10)],x=-2..2);

Gibbs phenomenon

f_step:= (x,N) -> 4/Pi*add(sin((2*n-1)*Pi*x)/(2*n-1),n=1..N):

plot(f_step(x,10),x=-2..2);

plot(f_step(x,400),x=0.8..1);