$\sum_{n=1}^\infty\dfrac{\cos (3^nx)}{2^n}$
$ \leq$ $ \sum_{n=1}^\infty\left|\dfrac{\cos (3^nx)}{2^n}\right|$ $ $
$\leq \sum_{n=1}^\infty\dfrac{1}{2^n} = \dfrac{1}{2}\sum_{k=0}^\infty\left(\dfrac{1}{2}\right)^k$
This is simply a geometric series with $|r| < 1$ which implies $\sum |a_n|$ Is convergent.
We all know that if $\sum |a_n|$ converges then so is $\sum a_n$. So option D is correct.