SOLUTION
The Infinite geometric series formula is:
[tex]\begin{gathered} S_{\infty}=\frac{a}{1-r} \\ \text{where a is the first term} \\ r\text{ is the common ratio} \end{gathered}[/tex]0.31313131..can be written in a series form as:
[tex]0.31+0.0031+0.000031+0.00000031+\text{.}\ldots[/tex]From this series,
[tex]\begin{gathered} a=0.31=\frac{31}{100} \\ r=\frac{0.0031}{0.31}=\frac{1}{100} \end{gathered}[/tex]Substituting these parameters into the infinite geometric series formula:
We will have:
[tex]\begin{gathered} \frac{\frac{31}{100}}{1-\frac{1}{100}} \\ \frac{\frac{31}{100}}{\frac{99}{100}} \\ \frac{31}{100}\times\frac{100}{99} \\ \frac{31}{99} \end{gathered}[/tex]The final answer is 31/99
