Question

In Exercises \(47-52,\) (a) write the repeating decimal as a geometric series and (b) write its sum as the ratio of two integers $$ 0 . \overline{4} $$

Short answer

The repeating decimal \(0.\overline{4}\) can be written as a rational number \( \frac{4}{9} \)

Step-by-step solution

Formulate as a Geometric Series

The periodic decimal \(0.\overline{4}\) can be written as \(0.44444...\) etc. This can be formulated as a geometric series where the first term, \(a_1 = 0.4\) and the common ratio, \(r = 0.1\) as every next term is obtained by multiplying the previous term by \(0.1.\) So our geometric series looks like this: \(0.4, 0.04, 0.004,...\)

Apply Geometric Series Sum Formula

The sum, \(S_1\) of an infinite geometric series, where the absolute value of the common ratio is less than 1, can be found by the formula \(S_1=\frac{a_1}{1 - r}\). Applying this formula to our series, we get \(S_1=\frac{0.4}{1 - 0.1}\)

Simplify the Expression

Simplifying the above expression \(S_1=\frac{0.4}{0.9}\), we find that \(S_1 = \frac{4}{9}\).