I can’t solve if $\;\sum_{k=1}^{\infty}\left(1-\cos(\frac{\pi}{k})\right)$ diverge or converge, would anyone mind helping me ?
I'm able to see that as $k \rightarrow \infty$ the $\,n^{\mathrm{th}}$ number will be $0$, therefore my guess would be converge, but I also know that it does not necessary converge when the $\,n^{\mathrm{th}}$ number is $0$.
I’m also able to observe that $a_k$ is decreasing and positive and therefore the integral criteria could be useful, and when integrating $\int_{1}^{\infty} 1 dx = \infty$, is this enough for saying that this sum diverge ?