Integral Sequence

Calculus Level 3

\[ \int_{0}^{i2\pi} \frac{e^{x}}{2(2-e^{x})} \, dx + \int_{0}^{i2\pi} \frac{e^{2x}}{4(2-e^{x})}\, dx+ \int_{0}^{i2\pi} \frac{e^{3x}}{8(2-e^{x})}\, dx +\cdots = \, ? \]

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

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