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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