I'm confused by the hint on the following exercise from Peter Olver's Introduction to Partial Differential Equations:
Suppose $a_k, b_k$ are the Fourier coefficients of the function $f(x)$. To which function does the Fourier series $$\frac{a_0}{2} + \sum_{k=0}^\infty [a_k \cos(2kx) + b_k \sin(2kx)] \enspace \text{converge?}$$ Hint: The answer is not $f(2x)$.
Does it refer to the fact that the period of the new series is $\pi$, so it's not the Fourier series of $f(2x)$ on the entire interval $[-\pi, \pi]$, but instead only on $[-\pi/2, \pi/2]$?