Separate and conquer

Geometry Level 5

\[\sum _{ n=1 }^{ \infty }{ \dfrac { \cos { \left( \frac { n \pi }{ 3 } \right) } }{ { 3 }^{ n } } } =\frac { \alpha }{ \beta }\] The equation above holds true for coprime positive integers \(\alpha\) and \(\beta\). Find \(\alpha+\beta\).

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