I have calculated the Fourier Series of $\cos(\frac{x}{2})$ as:
$$f(x) \sim \frac{2}{\pi} + \frac{4}{\pi} \sum_{n=0}^\infty \frac{(-1)^n}{1-4n^2} \cos(nx)$$
and the Fourier Series of $\cos(x)$ as:
$f(x) \sim \sum_{n=0}^\infty a_n \cos nx$ where $a_n = 1$ for $n= 1$ and $0$ otherwise.
My question is: Can you obtain the Fourier Series of $\cos(x)$ from the Fourier Series of $\cos(\frac{x}{2})$ by changing the variable from $x$ to $2x$?
I can't see how you can go from having all the $a_n$ with values to just the one with a value, just by changing $x$?