So wavy, when all I wanted was a straight line! I'd like a series that can approximate not just the values of a function, but all its derivatives as well.
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.