How to determine if this converges?

Question

I find this one to be hard, I tried expanding $\cos$ into a Taylor series, but that didn't work out well because I couldn't apply $p$-series... $$ \sum^{\infty}_{n=1} n (1-\cos(\pi/n)) $$

Answer 1

Hint

Use the inequality $1-\cos x\ge \frac{x^2}{3}$ for $-2\le x\le 2.$

Answer 2

We know that $$1-\cos(\frac{\pi}{n})=2\sin^2(\frac{\pi}{n})$$ also $$\sin^2(\frac{\pi}{n}) \sim (\frac{\pi}{n})^2, \ \ \ \ \text{as} \ \ \ n\to \infty$$ therefore $$\sum\limits_{n=1}^{\infty}n(1-\cos(\frac{\pi}{n}))=\sum\limits_{n=1}^{\infty}2n\sin^2(\frac{\pi}{n})\sim \sum\limits_{n=1}^{\infty}2n (\frac{\pi}{n})^2=\sum\limits_{n=1}^{\infty} 2\frac{\pi^2}{n}$$. We know the series $\sum\limits_{n=1}^{\infty} \frac{\pi^2}{n}$ diverges , So this series $\sum\limits_{n=1}^{\infty}n(1-\cos(\frac{\pi}{n}))$ will also diverge.