jmc

We have: $$0.1\overline{7}=\frac{1}{10}+\frac{7}{10^2}+\frac{7}{10^3}+\frac{7}{10^4}+\cdots .$$After the first term, the series on the right is an infinite geometric series with first term $7/10^2$ and common ratio $1/10$. So we have: $$0.1\overline{7}=\frac{1}{10}+\frac{\frac{7}{10^2}}{1-\frac{1}{10}}=\frac{1}{10}+\frac{\frac{7}{10^2}}{\frac{9}{10}}=\frac{1}{10}+\frac{7}{90}=\frac{9+7}{90}=\frac{16}{90}=\boxed{\frac{8}{45}}.$$