I am currently taking a Fourier Analysis module in my Physics course and I was asked this question as a homework task which has caused me a great deal of confusion.
This attached image shows how the truncated series of both the triangular and square waves converge. In essence, why is it that the triangular series converges much quicker than the square series?
Edit: This is the actual question I got for all those curious.
How do you want to look at it?
Because the square wave has discontinuities in the waveform where the triangle wave only has discontinuities in the derivative, and these discontinuities require more high harmonics to reach a close approximation of the original waveform.
Because the square wave has more energy in the high harmonics, cutting off the Fourier series at any given harmonic causes a bigger change in the square wave than in the triangle wave.