Binary expansions of $\frac{1}{2}$

Question

I was reading Elements of Set Theory by Herbert B. Enderton, and I saw there is written: $$0.1000...=0.0111...=\frac{1}{2}$$

I don't understand how $0.0111...$ is a binary expansion of $\frac{1}{2}$. I'm giving you the text of the book to see if I'm getting it wrong.

Answer 1

$0.01111\dots$ means the value of the series $$ \sum_{k=2}^\infty 2^{-k} $$ Do you know how to evaluate that series?