Example 12.1.5. Application: Converting repeating decimals to fractional form.
Consider this example:
\begin{align*}
2.48484848... & = 2 + 0.48 + 0.0048 + 0.000048 + ...\\
& = 2 + 0.48(1 + 0.01 + 0.0001 + ... ) = 2 + 0.48 \sum_{k=0}^\infty (0.01)^k
\end{align*}
Therefore, applying the Geometric Series
\begin{align*}
2.48484848... & = 2 + 0.48 \frac{1}{1-0.01} \\
& = 2 + 0.48 \frac{100}{99} = 2 + \frac{48}{99}
\end{align*}