Find the sum of the convergent series ∑ n = 3 ∞ ( cos ( π n ) − cos ( π n + 1 ) ) {\displaystyle \displaystyle \sum \limits _{n=3}^{\infty }{\bigg (}\!\cos {\Big (}{\frac {\pi }{n}}{\Big )}-\cos {\Big (}{\frac {\pi }{n+1}}{\Big )}{\bigg )}} . Simplify your answer completely.
