2015
Problem - 4563
Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$
The conclusion of # 4562 we have $1-\cos{x} < x^2$ for all $x > 0$. It follows that
$$0\le \sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right) < \sum_{n=1}^{\infty}\left(\frac{\pi}{n}\right)^2=\pi^2\sum_{n=1}^{\infty}\frac{1}{n^2}$$
Therefore, the given series converges.