Fourier series are -key- in math and physics (and music, etc.), and
can and should be introduced ASAP.
Also, they can be easily computed for polynomials using integration by
parts, which means they're also "easy" in some sense.

Note: for odd/even functions, the fourier series will have only
sin/cos (compare with Taylor series, where you'll have only odd/even
degree!)

Exercise:

The correct statement of Fourier Series is something like:
the functions $z \mapsto z^n$, for $n\in \bZ$ (i.e., the representations)
form a Hilbert basis for ``some space of functions'' $S^1 \to \bC$.
In real terms, the functions $\sin nx, \cos nx$ form a basis for
(periodic) functions on $[-\pi,\pi]$.
(the vagueness is because specifics involve subtle analysis:
 for what function space, i.e., what functions and what topology,
 do fourier series converge?
 Suffice to say that for continuous functions in the uniform metric
 or in the measure metric, we have convergence)

However, on a first pass students:
- won't know what a basis is
- won't be concerned with details of convergence

Instead, one can and should be informal:
One can introduce a basis as ``coordinates'':
then writing $f=$ (fourier series of $f$) is just like writing
$(1,1,1,\dots)=(1,0,\dots) + (0,1,0,\dots) + \dotsb$

I like introducing it by talking about spectrum and sound,
and timbre etc. -- it's totally applicable!

(include orthonormal stuff:
 $(1/\sqrt{2\pi})e^{k\pi i x}$ is the hermitian basis;
 over the reals you take
 $(1/\sqrt{2\pi})\sin k\pi i x$ (etc. with cosine)
 and hence fourier coefficients are:
\begin{align*}
   a_0 &= \frac{1}{2\pi} \int_{-\pi}^\pi f(x) \dx\\
   a_k &= \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos (kx)\dx\\
   b_k &= \frac{1}{\pi} \int_{-\pi}^\pi f(x) \sin (kx)\dx
\end{align*}
   Then
   $$f(x) = a_0 + \sum_{k=1}^\infty a_k \cos (kx) + b_k \sin (kx)$$
(the $1/2$ is b/c you're not adding a complex conjugate at $0$
 to get the real transform; otherwise you're getting twice the real part...)

