Modern computing uses binary representations of real numbers. Consider binary decimal

representations of the form $0.123...$where every digit $1,2,3,..$is either $0$ or $1$ The base

ten representation of the binary number $0.123...$ is the infinite series $1$

$2$

$1+$

$2$

$2$

$2+$

$3$

$2$

$3+$