Ratio of Complex Series

Algebra Level 3

\[\frac{1+\left(\dfrac{1}{\sqrt3}\right)i+\left(\dfrac{1}{3}\right)i^{2}+\left(\dfrac{1}{3\sqrt3}\right)i^{3}+\cdots}{1-\left(\dfrac{1}{\sqrt3}\right)i+\left(\dfrac{1}{3}\right)i^{2}-\left(\dfrac{1}{3\sqrt3}\right)i^{3}+\cdots} = \, ? \]

\[\] Clarification: \(i=\sqrt{-1}\).

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