Suma de $A\cos (\omega n+\phi)$

Question

Intento evaluar la siguiente suma:
Mi problema original es

$$\lim_{N \to \infty} \frac{1}{2N+1} \sum_{n=-N}^N \left|A \cos(\omega n+\phi)\right|^2$$

Ahora estoy atascado en el cálculo de la suma de la parte resaltada.
¡Cualquier ayuda!

Answer 1

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\begin{align} &\color{#f00}{\lim_{N \to \infty}\bracks{{1 \over 2N + 1}\sum_{n = -N}^{N} \verts{A\cos\pars{\omega n + \phi}}^{2}}} \\[3mm] = & \verts{A}^{2}\lim_{N \to \infty}\bracks{{1 \over 2N + 1}\sum_{n = -N}^{N} \verts{\cos\pars{\omega n}\cos\pars{\phi} - \sin\pars{\omega n}\sin\pars{\phi}}^{2}} \\[3mm] = & \verts{A}^{2}\lim_{N \to \infty}\braces{{1 \over 2N + 1}\sum_{n = -N}^{N} \bracks{\cos^{2}\pars{\omega n}\cos^{2}\pars{\phi} + \sin^{2}\pars{\omega n}\sin^{2}\pars{\phi}}} \\[3mm] = & \verts{A}^{2}\lim_{N \to \infty}\braces{{1 \over 2N + 1}\sum_{n = -N}^{N} \bracks{{1 + \cos\pars{2\omega n} \over 2}\cos^{2}\pars{\phi} + {1 - \cos\pars{2\omega n} \over 2}\sin^{2}\pars{\phi}}} \\[3mm] = & \half\,\verts{A}^{2}\lim_{N \to \infty}\braces{{1 \over 2N + 1}\sum_{n = -N}^{N} \bracks{1 + \cos\pars{2\phi}\cos\pars{2\omega n}}} \\[3mm] = & \half\,\verts{A}^{2}\bracks{1 + \cos\pars{2\phi} \lim_{N \to \infty}{1 \over 2N + 1}\sum_{n = -N}^{N}\cos\pars{2\omega n}} \\[3mm] = & \color{#f00}{\left\lbrace\begin{array}{lcl} \half\,\verts{A}^{2}\bracks{1 + \cos\pars{2\phi}} & \mbox{if} & \omega \equiv \ell\pi\,,\quad\ell \in \mathbb{Z} \\ \half\,\verts{A}^{2} && \mbox{otherwise} \end{array}\right.} \end{align}