jmc

This is a finite geometric series with first term $\frac{1}{3}$ and common ratio $\frac{1}{3}$. There are five terms, so the sum of this series is $\frac{\frac{1}{3}\left(1-{\left(\frac{1}{3}\right)}^5\right)}{1-\frac{1}{3}}=\boxed{\frac{121}{243}}$.