How to determine if this series converges?

Question

Does this series converge? I tried using limit comparison, and I don't know what to try next...

$$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$

Answer 1

You can prove by various methods that if $x$ is small then $$1-\cos x<\frac{x^2}{2}\ .$$ So if $n$ is large then $$1-\cos\Bigl(\frac\pi n\Bigr)<\Bigl(\frac{\pi^2}2\Bigr)\frac1{n^2}\ ,$$ and you can use a comparison test.

Answer 2

You may use the Maclaurin expansion for $\cos x$: $$ \cos (x) = 1 -\frac{x^2}{2}+ \mathcal{O}(x^4) $$ giving $$ \sum_{n=2}^{\infty}\left( 1-\cos \left(\frac{\pi}{n}\right)\right)=\sum_{n=2}^{\infty}\frac{\pi^2}{2n^2}+\sum_{n=2}^{\infty}\mathcal{O}(\frac{1}{n^4}) $$ concluding that your initial series converges.

Answer 3

Note that $$ \sum_{n=1}^{\infty}1-\cos\left(\pi/n\right)\leq\int_0^\infty 1-\cos\left(\pi/x\right)dx=\pi^{2}/2 $$