Cantor Set Looking for what is Not remaining?

Question

I have the Cantor Set parsed out into as follows: \begin{align}\mathrm{Segment\ 1}&=[0,\frac13],[\frac23,1]\\ S_1&=\frac13\\ \mathrm{Segment\ 2}&= [0,\frac19],[\frac29,\frac13],[\frac23,\frac79],[\frac89,1]\\ S_2&=\frac13+\frac29\end{align} My question are my $S_n$s what is not remaining? Because I have to construct a series of what is not remaining so is that right because then so far I have the following. \begin{equation}\sum_0^\infty \frac{1}{3}(\frac23)^n=\frac13+\frac29+\frac{4}{27}+... \end{equation}

Answer 1

I'm not sure if I understood you correctly. At each step you're calculating a measure of the complement of the Cantor set (in the unit interval). Since the Cantor set is the subset of $[0,1]$ then $\mu(\mathcal{C})=\mu([0,1])-\mu([0,1]\setminus\mathcal{C})$. Now just observe that at step $n$ you've removed total length of $\sum_{k=1}^n\frac{2^{n-1}}{3^n}$ which is geometric series that converges to $1$.