jmc

$$0.72\overline{6}=\frac{7}{10}+\frac{2}{10^2}+\frac{6}{10^3}+\frac{6}{10^4}+\frac{6}{10^5}+\cdots .$$After the first two terms, the series on the right is an infinite geometric series with first term $6/10^3$ and common ratio $1/10$. So, we have $$0.72\overline{6}=\frac{7}{10}+\frac{2}{10^2}+\frac{\frac{6}{10^3}}{1-\frac{1}{10}}=\frac{72}{100}+\frac{6}{900}=\frac{654}{900}=\boxed{\frac{109}{150}}.$$