Change of variable in Fourier series

Question

I am not finding any document online discussing how changing the variable affects the Fourier series. Any thought on the rule ?

Answer 1

Hint:

The series for $f(2x)$ is identical to the new series, so they must be the same over some interval.

To resolve this seemingly paradoxical result, ask yourself:

What's the period of the original series? What can you say about the period of the new series?