Answer
Converges
Work Step by Step
We can see that $\Sigma_{k=1}^{\infty} \dfrac{4+|cos x|}{k^3} \lt \Sigma_{k=1}^{\infty} \dfrac{6}{k^3}$
But the series $\Sigma_{k=1}^{\infty} \dfrac{6}{k^3}$ is a p-series with $p=3$, so it converges by the p-series. This implies that the given series is less than a converging series, so it converges.
