Given a function $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$. Peaks are at odd $X$ integer values.
How to find the scaling with $N$ of the FWHM (full width half max) and peak max? For example for the peak at $X =1$.
The maxima seems to scale as $2N+1$ applying a limit (L'Hopital rule)?
$$1 + \cos \theta = 2 \cos^2 \frac{\theta}{2},$$ so the function becomes $$f_n(x) = \left( \frac{\cos \frac{(2n+1)\pi x}{2}}{\cos \frac{\pi x}{2}} \right)^2.$$ Now performing L'Hopital's rule on the interior function trivially yields $$\lim_{x \to 1} f_n(x) = (2n+1)^2.$$
As for the FWHM, this is difficult to compute and does not have an elementary closed form when $n > 2$. The value of $x \in (0,1)$ for which $f_n(x) = (2n+1)^2/2$ (very) approximately behaves as $1 - c/n$ for a suitable constant $c$.
Explicit computation of the limit: Consider $$\lim_{x \to \pi/2} \frac{\cos (2n+1)x}{\cos x} = \lim_{x \to \pi/2} \frac{-(2n+1) \sin (2n+1)x}{-\sin x} = 2n+1.$$ Therefore, $$\lim_{x \to 1} f_n(x) = (2n+1)^2$$ as claimed.