$\begin{aligned} & \frac{(1+i)^{2011}}{(1-i)^{2009}}=\frac{(\sqrt{2})^{2011}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{2011}}{(\sqrt{2})^{2009}\left(\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}\right)^{2009}} \\ & =\frac{(\sqrt{2})^2\left(e^{\frac{i \pi}{4}}\right)^{2011}}{\left(e^{-\frac{i \pi}{4}}\right)^{2009}}=2\left(e^{\frac{i \pi}{4}}\right)^{4020} \\ & =2\left(e^{i \pi}\right)^{1005}=2(-1)^{1005}=-2\end{aligned}$