What is 'degrees of freedom' when using Fourier series to express a periodic waveform?

We can express any desired periodic waveform using Fourier series.

In the book I am studying from it’s said:

‘We see that with Fourier series, we can produce any desired periodic waveform and extract its wave number content (via the $a_n$ and $b_n$). But even though we have used an infinite number of plane wave components, we still evidently do not have enough “degrees of freedom” to produce a truly localized wave packet."

I was wondering what the ‘degrees of freedom’ in this context mean.

For my understanding a wave packet is:

The most general solution of the wave equation can be shown to be given by any (suitably differentiable) function of the form $phi(x,t)=f(xpm vt)$ since it satisfies:

$$(pm v)^2 ?”(?+vt)=frac{partial^2phi(x,t)}{partial t^2}=v^2 frac{partial^2 phi(x,t)}{partial x^2}=v^2 f”(x pm vt).$$

This implies that any appropriate initial waveform, $f(x)$, can be turned into solution of $frac{partial^2phi(x,t)}{partial t^2}=v^2 frac{partial^2 phi(x,t)}{partial x^2},$ $f”(x pm vt),$ which propagates to the right (-) or left (+) with no change in shape.
Such solution is called wave packet.