We get it.

You love math, but you're also short on time.

You want to learn what that Fourier series thing is about, but you don't don't want to go through 2 semesters of differential equations.

So here it is:

The idea of the Fourier series representation of a function is that you can represent a periodic function with period $\pi$ as the sum of a bunch of sine and cosine functions (plus a constant).

In Greek symbols:

$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n\text{sin}(x) + b_n\text{cos}(x)$$

But what are those $a_0$,$a_n$, and $b_n$ coefficients?

$$a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)dx}$$
$$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)\text{sin}(x)dx}$$
$$b_n = \frac{1}{\pi} \int_{-\pi}^{\pi}{f(x)\text{cos}(x)dx}$$

And there you have it– The Fourier series in (hopefully) 20 seconds.