More on fourier series, see following comment.

Question

anonymous · Answer

Just want to check my answer, mathematica is taking too long to plot this sum at an appropriate alpha value.

Find the Fourier series of 
$$f(x)=\cos(\alpha x)$$ Where alpha is not an integer.


For your convenience:

$$f=\frac{a_0}{2}+\sum_{n=1}^{\infty}[a_ncos(nx)+b_nsin(nx)]$$$$a_0=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)dx$$$$a_n=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)cos(nx)dx$$$$b_n=\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x)sin(nx)dx$$

My answer is:
$$\frac{\sin(\alpha \pi)}{\alpha \pi}+\sum_{n=1}^{\infty}(\frac{\sin((\alpha-n)\pi)}{\pi(\alpha-n)}+\frac{\sin((\alpha+n)\pi)}{\pi(\alpha+n)})\cos(nx)$$