I want to know whether there are a finite number of coefficients in a Fourier series of a periodic function (with period $P$), whose magnitude are above a certain threshold. Those coefficients can can be calculated with,
$$ a_n = \frac{2}{P}\int_{0}^{P}{f(x)\cos\left(\frac{2\pi nx}{P}\right)dx}, $$
$$ b_n = \frac{2}{P}\int_{0}^{P}{f(x)\sin\left(\frac{2\pi nx}{P}\right)dx}. $$
For most functions $a_n$ and $b_n$ will go to zero when $n$ goes to infinity, such as the series of the square and sawtooth wave. Thus those functions would satisfy the requirement.
An example of a function which does not satisfy this would be,
$$ f(x)=\sum_{n=0}^\infty \cos\left(\frac{2\pi nx}{P}\right), $$
which seems to be a periodic Dirac delta function (correct me if I am wrong). Is there an easy way to identify whether a function wouldn't satisfy the requirement. I suspect that this would happen for functions which have periodic limits where the function goes to $\pm\infty$. But I do not know how to prove this and whether this would include all functions.
A side question, am I correct that only periodic functions that are continuously differentiable will have a finite number of coefficients which are non-zero?
I feel obligated to give the mathematician's answer:
A Fourier series whose coefficients do not go to 0 does not converge. Also, an example of a continuously differentiable periodic function with infinitely many nonzero Fourier coefficients whose Fourier series converges is $\sin \sin x$.