Question 4
Consider the real sequence $\{x_n\}_{n=1}^{+\infty}$ defined by
$$x_1=1,\qquad x_{n+1}=\frac{(2+2\cos2\alpha)x_n+\cos^2\alpha}{(2-2\cos2\alpha)x_n+2-\cos2\alpha}\quad(n=1,2,\ldots),$$
where $\alpha$ is a real parameter. Determine all possible values of $\alpha$ such that the sequence $\{y_n\}_{n=1}^{+\infty}$ with
$$y_n=\sum_{k=1}^{n}\frac1{1+2x_n}$$
has a finite limit as $n$ tends to infinity, and determine those limits.
$$x_1=1,\qquad x_{n+1}=\frac{(2+2\cos2\alpha)x_n+\cos^2\alpha}{(2-2\cos2\alpha)x_n+2-\cos2\alpha}\quad(n=1,2,\ldots),$$
where $\alpha$ is a real parameter. Determine all possible values of $\alpha$ such that the sequence $\{y_n\}_{n=1}^{+\infty}$ with
$$y_n=\sum_{k=1}^{n}\frac1{1+2x_n}$$
has a finite limit as $n$ tends to infinity, and determine those limits.