jmc

This is a finite geometric series with first term $\frac{1}{3}$, common ratio $\frac{1}{3}$ and $6$ terms. Therefore the sum is: $$\frac{\frac{1}{3}\left(1-\frac{1}{3^6}\right)}{1-\frac{1}{3}}=\frac{\frac{3^6-1}{3^7}}{\frac{2}{3}}=\frac{3^6-1}{2\cdot 3^6}=\frac{729-1}{2\cdot 729}=\boxed{\frac{364}{729}}.$$