Is this series convergent or diveregent...and from what method I can prove this.
$\displaystyle
\sum_{n=1}^{\infty} \frac{cos (n)}{n}
$
To prove convergence, use Dirichlet's test.
If you want to know the sum of the series, notice that $\displaystyle \sum_{n=1}^\infty\frac{z^n}n = \log(1-z)$. If $\displaystyle z=e^{i\theta}$ then $\displaystyle \log(1-z) = \ln(2\sin\tfrac\theta2) + i(\theta+\pi)/2$. Take the real part to see that $\displaystyle \sum_{n=1}^\infty\frac{\cos n\theta}n = \ln(2\sin\tfrac\theta2)$. In particular, if $\displaystyle \theta=1$ then $\displaystyle \sum_{n=1}^\infty\frac{\cos n}n = \ln(2\sin\tfrac12)$. That argument is not rigorous, because the series for $\displaystyle \log(1-z)$ has radius of convergence 1, so you cannot assume that it behaves well when |z| = 1. However, once you have used the Dirichlet test to prove convergence, the rest of the argument works to give the correct result.