Does the Sum Converge?

Calculus Level 3

Let \(\theta_k\) be a randomly generated real number which varies with \(k\). Does the following sum converge?

\[\Large \sum^\infty_{k=0}\left(\frac{4}{3}\right)^{k} \, e^{(-\frac{k}{3} + i \theta_k)}\]

\[\] Notation: \(i=\sqrt{-1}\) denotes the imaginary unit.

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