# Trigonometric Recurrence and its Limit!

Geometry Level 5

$\large{ x_{n+1} = \dfrac{(2 + \cos(2\alpha))x_n + \cos^2(\alpha)}{(2 - 2\cos(2\alpha))x_n + 2 - \cos(2\alpha)} }$

Consider the sequence of real numbers $$(x_n)_{n=1}^\infty$$ defined as above with $$x_1 = 1$$ for every $$n \in \mathbb N$$, where $$\alpha$$ is a real parameter.

$$\forall \, n \in \mathbb N$$, let $$\large{y_n = \displaystyle \sum_{k=1}^n \dfrac{1}{2x_k +1}}$$

For specific values of $$\alpha$$, if the sequence $$(y_n)_{n=1}^\infty$$ has a finite limit, find that limit.

Bonus: Determine the values of $$\alpha$$ for which the sequence $$(y_n)_{n=1}^\infty$$ has a finite limit.

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