How to write $\frac{3}{10}$ in ternary expression for Cantor?

Question

I try to show that $\frac{3}{10}$ belong to Cantor set but it is not an end point. For that I try to show the ternary expression of $\frac{3}{10}$. Is there any way else to prove this?

Answer 1

$$\frac{3}{3^2+1}=\frac{24}{3^4-1}=24\cdot\left(\frac{1}{3^4}+\frac{1}{3^8}+\frac{1}{3^{12}}+\ldots\right)$$ hence the ternary representation of $\frac{3}{10}$ is given by $$ 220_3 \cdot 0.\overline{0001}_3 = \color{red}{0.\overline{0220}}_3.$$