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

Since the series $\sum _{n=1}^{\mathrm{\infty }}\cos ({\theta }_n)$ converges, it must be that the terms of the sum converge to 0: $$\underset{n\to \mathrm{\infty }}{\lim }\cos ({\theta }_n)=0.$$ 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 $\frac{\pi }{2}$.