Answer
Absolutely convergent.
Work Step by Step
Given $$\sum_{n=1}^{\infty} \frac{\sin 4 n}{4^{n}}$$
Since
$$ \left| \frac{\sin 4 n}{4^{n}}\right|\leq \frac{1}{4^{n}} $$
Then by using the comparison test with the convergent series $ \displaystyle\sum_{n=1}^{\infty}\frac{1}{4^{n}}$ ( geometric series with $|r|<1$), we know that $\displaystyle\sum_{n=1}^{\infty} \frac{\sin 4 n}{4^{n}}$ is absolutely convergent.