$\sum_\limits{n=1}^{\infty}(1-\cos(\frac{\pi}{n}))$ convergence proof

Question

Check if the following series converge or diverge:$\sum_\limits{n=1}^{\infty}(1-\cos(\frac{\pi}{n}))$

I have tried the integral test since the series are decreasing to zero as $n\to\infty$, but $\int_1^\infty 1-\cos(\frac{\pi}{n}) dn=n-\sin(\frac{\pi}{n})|_0^\infty$, which diverges. I am not seeing what test could I applied that would deliver me the desired result convergence, since the book solution states that the series converge.

Question:

What do you think of the series? What test shall I use?

Thanks in advance!

Answer 1

Hint: Use the fact that$$\lim_{n\to\infty}\frac{1-\cos\left(\frac\pi n\right)}{\frac1{n^2}}\in(0,+\infty).$$