> # FOURIER SINE SERIES OF A SQUARE WAVE

>

> # f(x) = 1 on (0,Pi); f(x) = -1 on (-Pi,0); f has period 2*Pi.

>

> # First we need to calculate the Fourier coefficients.

> bn := (2/Pi)*Int(sin(n*x), x=0..Pi);

> value(%);

> # This is 0 unless n is odd.

>

> # Note that, unlike the coefficients of the triangle wave, these decrease like 1/n, not 1/n^2.

>

> bodd := k -> (4/Pi)*(1/(2*k+1));

> partialsum := K -> sum(bodd(k)*sin((2*k+1)*x), k=0..K);

> plot(partialsum(0), x=-8..8);

> plot(partialsum(1), x=-8..8);

> plot(partialsum(2), x=-8..8);

> plot(partialsum(4), x=-8..8);

> plot(partialsum(8), x=-8..8);

> plot(partialsum(16), x=-8..8);

> # Let's look up close.

> plot(partialsum(16), x=-1..4);

> plot(partialsum(32), x=-1..4);

> plot(partialsum(64), x=-1..4);

>