Answer
The series $\Sigma_{n=1}^{\infty}\frac{\cos n}{2^n}$ converges absolutley.
Work Step by Step
Since $\cos n\leq 1$ then we have $\frac{\cos n}{2^n}\leq \frac{1}{2^n}$. Now, the series
$\Sigma_{n=1}^{\infty}\frac{1}{2^n}$ is a geometric series with $r=1/2\lt1$, which converges. Hence, the positive series $\Sigma_{n=1}^{\infty}|\frac{\cos n}{2^n}|$ converges. Thus, the series $\Sigma_{n=1}^{\infty}\frac{\cos n}{2^n}$ converges absolutley.
