Converting repeating decimal in base b to a fraction the same base The repeating decimal .36666... in base 8 can be written in a fraction in base 8.
I understand simple patterns such as 1/9 in base 10 is .1111.... so 1/7 in base 8 is .1111.
But I'm not too sure how to convert this decimal in this base to the fraction in the same base.  

\begin{align}
0.3\bar{6}_8
&= \frac{3}{8} + 6\left(\frac{1}{8^2}+\frac{1}{8^3} + \cdots\right)\\
&= \frac{3}{8} + \frac{6}{8^2}\left(1+\frac{1}{8}+  \frac{1}{8^2} +\cdots\right)\\
&= \frac{3}{8} + \frac{6}{8^2}\frac{1}{1-(1/8)} & \text{geometric series}\\
&= \frac{3}{8} + \frac{3}{28}\\
&= \frac{27}{56}\\
&= \frac{33_8}{70_8}.
\end{align}
