Chapter 10.3, Problem 88E

Binary numbers Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0. b₁b₂b₃… where each of the digits b₁, b₂, b₃, ... is either 0 or 1. The base-10 representation of the binary number 0. b₁b₂b₃… is the infinite series $\frac{b_1}{2^1}+\frac{b_2}{2^2}+\frac{b_3}{2^3}+\cdots$.

88. Verify that the base-10 representation of the binary number 0.010101... (which can also be written as 0. $\bar{0}\bar{1}$) is $\frac{1}{3}$.