Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 12/Solution 1

Since the series $\sum _{n=1}^{\infty }\cos(\theta _{n})$ converges, it must be that the terms of the sum converge to 0:

$$\lim _{n\to \infty }\cos(\theta _{n})=0.$$

This is a consequence of Theorem 3.3.1 in CLP . Note in particular that it does not matter what precise value the series converges to.

We see then that, for large $n$, the number $\cos(\theta _{n})$ is approximately $0$. Since $\cos$ is a continuous function, it follows that $\theta _{n}\approx {\frac {\pi }{2}}+k\pi$, where $k$ is some integer. But the only integer $k$ for which ${\frac {\pi }{2}}+k\pi$ is in the range $(0,3)$ is 0.

Therefore the sequence $\theta _{n}$ converges to ${\frac {\pi }{2}}$.