MathDB

The sequence (x_n)^{\infty}_{n\equal{}1} is defined by x_1 \equal{} 1 and x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}, for all

n \in\mathbb{N}

, where

\alpha

is a given real parameter. Find all values of

\alpha

for which the sequence

(y_n)

given by y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1} has a finite limit when n \to \plus{}\infty and find that limit.

Problem Statement