Finding a Laurent series for a complex function with two poles where one is outside of the region

complex-analysis laurent-series

The function is analytic in $|z| <1$ and its power series expansion is its Laurent series also. So the correct answer is $\sum (1-2^{-n-1})z^{n}$ [You missed $z^{n}$ in the series]. Note that poles and risidues play no role in this.

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