Chapter 11 - Infinite Series - 11.2 Summing and Infinite Series - Exercises - Page 548: 39

$$\frac{31}{99}$$

Let $$ x=0 . \overline{31}=0.313131 \ldots $$ Then \begin{aligned} 100 x-x &=31.313131\cdots \cdots-0.313131\cdots \\ 99 x &=31\\ x &=\frac{31}{99} \end{aligned}