Rewriting 0.333... as 0.3 + 0.03 + 0.003 + ... gives a geometric series with first term 0.3 and ratio 0.1. What fraction does it sum to?

1. The ratio 0.1 lies between -1 and 1, so the infinite sum exists. 2. Apply S = a_1/(1 - r) with a_1 = 0.3 and r = 0.1. 3. The denominator is 1 - 0.1 = 0.9, so S = 0.3/0.9. 4. That simplifies to 1/3. 5. Reporting 3/10 would give only the first term of the series rather than its total. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.4 Series and Their Notations_