Why would you expand a square wave in a Fourier series? The periodic square wave
$$ f(x) = \cases{ 1 \text{ if } 0 \le x \le \pi \\ 0 \text { if } -\pi \le x < 0} $$
seems easy enough to work with. Why transform it into a series of sines and cosines?
I'm writing up take-home bonus problems for my Calculus II class. I know one of my students is interested in Fourier series and it's relevant to many others, I'm sure, so one of the problems is a easy Fourier expansion of the square wave. I'd like to include in the introduction why someone might turn a seemingly-simple function like this into an infinite sum of trig functions.

There are certainly many places within mathematics where the Fourier series / transform would be useful. I'd also like to mention that it's a huge component of many practical areas; one in particular is digital signal processing. The square wave you mentioned can literally be heard as a sound played out over time (i.e. signal represented in the time domain). What if you want to store that sound on a computer? And can you do it without using too much space? Computing the Fourier series / transform for the square wave lets you break it down into a set of frequencies (i.e. signal represented in the frequency domain). This leads to interesting ways to store and compress audio information. (A very simple example: viewing a signal in the frequency domain makes it easy to chop off frequencies inaudible to the human ear, reducing file size possibly at the cost of quality.)
You might be interested in:
Shannon-Nyquist Sampling Theorem
MP3 Encoding
