There is a technique to compute Fourier series much quickly, but I doubt how general this technique can be. Let's look at a simple example to see how the technique goes.
Compute Fourier series of the $2\pi$ periodic function $\dfrac{2\sin\theta}{5-4\cos\theta}$:
Note(if we are gifted enough ^_^) that $\dfrac{2\sin\theta}{5-4\cos\theta}$ is a special case of the function $v(r,\theta) = \dfrac{r\sin\theta}{1-2r\cos\theta+r^2}$ when $r = \frac{1}{2}$.
And $v(r,\theta)$ is the imaginary part of the complex function $f(z) = \dfrac{1}{1-z}$ with $z = re^{i\theta}$.
Expanding $f(z) = \dfrac{1}{1-z} = 1 + z + z^2 + z^3 + \cdots$, substituting $z = re^{i\theta}$, and pick up the imaginary part, we have $v(r,\theta) = r\sin\theta + r^2\sin 2\theta + r^3\sin 3\theta +\cdots$.Letting $r = \frac{1}{2}$, we derive the Fourier series of $v(\frac{1}{2},\theta)=\dfrac{2\sin\theta}{5-4\cos\theta}$, i.e.
$\tilde{v}(\frac{1}{2},\theta)=\frac{1}{2}\sin\theta + \frac{1}{2^2}\sin 2\theta + \frac{1}{2^3}\sin 3\theta + \cdots$.
I wonder whether the above approach can be applied to compute any or most $2\pi$ periodic function's Fourier series ? If not, what conditions should be put on the function to guarantee success of the above approach ? (If necessary, we may suppose all series we meet are convergent.)