Does the Series $\sum_{n=1}^{\infty} (1-\cos\frac{\pi}{n})$ Converge? 
Does the serie $\displaystyle\sum_{n=1}^{\infty}\Bigl(1-\cos\frac{\pi}{n}\Bigr)$ converge?

Limit test of $1-\cos\frac{\pi}{n}$
$$
\lim_{x\to\infty} 1-\cos\frac{\pi}{n} =
1 -\lim_{n\to\infty}\cos\frac{\pi}{n}= 1-1 = 0
$$
I've checked the necessary condition that $\lim_{n\to\infty} 1-\cos\frac{\pi}{n} = 0 $
But how do I check if $\sum_{n=1}^{\infty}(1-\cos\frac{\pi}{n})$ converges?

You can use L'hopital's rule (twice) to show that
$$\lim_{x \rightarrow 0} {1 - \cos x \over x^2} = {1 \over 2}$$
So you can use the limit comparison test with the series ${\displaystyle \sum_{n = 1}^{\infty} {1 \over n^2}}$ to show your series converges.
