Problem 46 - College Algebra

Question

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots$$

Step-by-Step Solution

Answer

The repeating decimal \$0.\overline{1}\$ can be written as the fraction \$1/9\$, in its lowest terms.

Step 1: Express the Number as an Infinite Geometric Series

The decimal \(0.\overline{1}\) can be written as an infinite geometric series: \(0.1 + 0.01 + 0.001 + 0.0001 + \cdots\). This can be put in the general form of a geometric series where the first term \(a = 0.1\) and the common ratio \(r = 1/10\).

Step 2: Use the Formula for the Sum of an Infinite Geometric Series

The sum \(S\) of an infinite geometric series can be found using the formula \(S = a / (1 - r)\), where `a` is the first term and `r` is the common ratio. This gives us \(S = 0.1 / (1 - 1/10)\).

Step 3: Simplify the Expression

First simplify the denominator of the fraction in the sum formula: \(1 - 1/10 = 0.9\). Then we can proceed with the fractional division: \(S = 0.1 / 0.9\). This will give us \(S = 1 / 9\).

Step 4: Check if the Fraction is in Lowest Terms

Check if the fraction \(1/9\) is in its simplest form. If the numerator and the denominator don't share any common factor other than 1, the fraction is in its simplest form. In this case, 1 and 9 don't have any common factor, hence \(1/9\) is the fraction in lowest terms.