For smooth functions, the Fourier series also approximates the derivative. The waviness you're seeing here is due to the Gibbs' phenomenon, since you're approxi

For smooth functions, the Fourier series also approximates the derivative. The waviness you're seeing here is due to the Gibbs' phenomenon, since you're approximating a discontinuous function. This is not a very tragic loss, since the derivative does not even exist where the waviness is the worst.