6. Find the sum $\frac{\sin \frac{\pi}{3}}{2}+\frac{\sin \frac{2 \pi}{3}}{2^{2}}+\cdots+\frac{\sin \frac{2010 \pi}{3}}{2^{2010}}$.
Answer: $\frac{\sqrt{3}}{3}\left(1-\frac{1}{2^{2010}}\right)$.

Criteria. "士" In the correct solution, the sums of finite geometric progressions are approximated by the sums of the corresponding infinite geometric progressions (and the answer $\approx \frac{\sqrt{3}}{3}$ is obtained).