Separate and conquer

Geometry Level 5

n=1cos(nπ3)3n=αβ\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 α\alpha and β\beta. Find α+β\alpha+\beta.

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