<s>[INST] Show that a number belongs or does not belong to a Cantor set I have trouble showing that $\frac{10}{31}$ and $\frac{45}{69}$ belong/do not belong to the Cantor set. I tried to do the following
$$\sum_{n = x}^\infty \frac{2}{3^n} = \frac{2}{3^x} + \frac{2}{3^{(x + 1)}} + \frac{2}{3^{(x + 2)}} + \dots = \frac{2}{3^x} \left[ 1 + \frac{1}{3^1} + \frac{1}{3^2} + \dots \right] = \frac{2}{3^x} \sum_{k = 0}^\infty \frac{1}{x^k} =  \frac{2}{3^x} \times \frac{1}{1- \frac{1}{3}} = \frac{1}{3^{x- 1}}$$
and then plug different values of $x = 1, 2, \dots$, but I could not seem to get these numbers. Does it mean they do not belong to the Cantor set? Please help if you can. Thank you so much.
 [/INST] Neither one is in the Cantor set. $\dfrac{10}{31} = 0.02220101110012..._{3}$, and $\dfrac{45}{69} = \dfrac{15}{23} = 0.12212110220122..._{3}$. Both has $1$ in their ternary presentation.
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