$\sum_{n=1}^{\infty } \left ( 1 - \cos(\frac{1}{n}) \right )$ converges or diverges? $$\sum_{n=1}^{\infty } \left ( 1 - \cos\left(\frac{1}{n}\right) \right )$$
I tried all of the tests but I couldnt solve this.Does it converges or diverges?

Hint: $1-\cos{(1/n)} = 2 \sin^2{(1/(2 n))}$.  How does $\sin{x}$ behave as $x \rightarrow 0$?
Once you have that, can you think of a sum that behaves the same as $n \rightarrow \infty$?  Does it converge or diverge?
