I'm trying to simplify the function $\cos\left(\frac{2\pi n}{3}\right)$ for integer $n$ into some sort of alternating series expression. For reference, I know that $\cos(\pi n)$ can be written as: $$ \cos(\pi n) = (-1)^n, \quad n\in\mathbb{Z}. $$
And when graphing $\cos\left(\dfrac{2\pi n }{3}\right)$, it can be clearly seen that its value alternates between $1$ and $-\frac{1}{2}$, with two occurrences of -$\frac{1}{2}$ for every occurrence of $1$.
How can I simplify this, if it is even possible? Also, if anyone can point me to somewhere I can read more about this topic, I would very much be interested.