1. **State the problem:** Express the repeating decimal $0.21\overline{21}$ as a fraction using the sum of an infinite geometric series. 2. **Recall the formula for the sum of an infinite geometric series:** $$S = \frac{a}{1 - r}$$ where $a$ is the first term and $r$ is the common ratio with $|r| < 1$. 3. **Rewrite the decimal as a sum:** $0.21\overline{21} = 0.21 + 0.0021 + 0.000021 + \cdots$ 4. **Identify the first term and common ratio:** - First term $a = 0.21$ - Common ratio $r = 0.01$ (each term is multiplied by $\frac{1}{100}$ to get the next) 5. **Apply the sum formula:** $$S = \frac{0.21}{1 - 0.01} = \frac{0.21}{0.99}$$ 6. **Convert to fraction:** $$\frac{0.21}{0.99} = \frac{\frac{21}{100}}{\frac{99}{100}} = \frac{21}{100} \times \frac{100}{99} = \frac{21}{99}$$ 7. **Simplify the fraction:** $$\frac{21}{99} = \frac{\cancel{3} \times 7}{\cancel{3} \times 33} = \frac{7}{33}$$ **Final answer:** $$0.21\overline{21} = \frac{7}{33}$$