Answer
The series is absolutely convergent.
Work Step by Step
Ratio Test: $\lim\limits_{n \to \infty}|\dfrac{a_{n+1}}{a_{n}}|=\lim\limits_{n \to \infty}|\dfrac{\frac{cos ((n+1) \pi/3)}{(n+1)!}}{\frac{cos (n \pi/3)}{n!}}|$
$=0\lt 1$
The series is absolutely convergent.