Question

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{1}=0.111 \ldots$$

Short answer

Answer: The repeating decimal 0.111... can be represented as the fraction 1/9.

Step-by-step solution

1. Write the repeating decimal as a geometric series

To represent the given repeating decimal \(0.\overline{1}\) as a geometric series, we can express it as an infinite sum of terms with a common ratio of 0.1. This is because each term in the decimal representation is 1 moved one place to the right, which is the same as multiplying by 0.1. Therefore, we can write the repeating decimal as follows: $$0.\overline{1} = 0.1 + 0.01 + 0.001 + \ldots$$ Now, we have the geometric series with the first term \(a_1 = 0.1\) and a common ratio \(r = 0.1\).

2. Use the sum formula for an infinite geometric series to find the sum

For an infinite geometric series with a common ratio \(|r| < 1\), the sum can be found using the formula: $$S = \frac{a_1}{1 - r}$$ In our case, the first term \(a_1 = 0.1\) and the common ratio \(r = 0.1\). Plugging these values into the formula, we get: $$S = \frac{0.1}{1 - 0.1} = \frac{0.1}{0.9}$$

3. Convert the decimal into a fraction

Now, we have the sum of the geometric series as a decimal: $$S = \frac{0.1}{0.9}$$ To convert the decimal into a fraction (a ratio of two integers), we can multiply the numerator and the denominator by 10 to eliminate the decimals: $$S = \frac{0.1 \times 10}{0.9 \times 10} = \frac{1}{9}$$ So, the repeating decimal \(0.\overline{1}\) can be represented as the fraction \(\frac{1}{9}\).