Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 15-20, write the given decimal as an infinite series, then find the sum of the series, and finally, use the result to write the decimal as a ratio of two integers (see Example 2). $$ 0.013013013 \ldots $$

Step-by-Step Solution

Answer

\(0.013013013\ldots = \frac{13}{999}\). The repeating decimal is \(\frac{13}{999}\).

Step 1: Recognize the Pattern

The given decimal \(0.013013013\ldots\) is a repeating decimal with the pattern \(013\). Every three digits, the same sequence repeats.

Step 2: Express as an Infinite Series

This repeating decimal can be expressed as an infinite geometric series: \[0.013013013\ldots = 0.013 + 0.000013 + 0.000000013 + \ldots\] Each following term is multiplied by \(0.001\) (or \(10^{-3}\)) with respect to the previous term.

Step 3: Write the Formula for the Series

The series from Step 2 can be written as: \[S = 0.013 + 0.013 \times 10^{-3} + 0.013 \times 10^{-6} + \ldots\] This is an infinite geometric series with first term \(a = 0.013\) and common ratio \(r = 0.001\).

Step 4: Calculate the Sum of the Series

To find the sum of an infinite geometric series, use the formula: \[S = \frac{a}{1 - r}\] where \(a = 0.013\) and \(r = 0.001\). Substitute these values: \[S = \frac{0.013}{1 - 0.001} = \frac{0.013}{0.999}\] Convert \(0.013\) and \(0.999\) to fractions: \[S = \frac{13}{1000} \div \frac{999}{1000} = \frac{13}{999}\] The sum of the series is \(\frac{13}{999}\).

Step 5: Write the Decimal as a Ratio

From Step 4, the sum of the series, which represents the decimal, is \(\frac{13}{999}\). Thus, the original decimal \(0.013013013\ldots\) can be written as the ratio \(\frac{13}{999}\).

Key Concepts

Repeating Decimals

Repeating decimals are numbers that have a sequence of digits that repeat infinitely. For example, in the decimal 0.013013013... the digits "013" continue indefinitely. Understanding repeating decimals is key because it allows us to represent them as fractions, which are easier to work with in mathematical equations and calculations.
When faced with a repeating decimal, our goal is often to express it as a fraction. This can be tricky because the decimal never ends, yet it has a regular repeating pattern. Recognizing and recording this pattern accurately is the first step in converting the decimal into a useful mathematical form.

Series Summation

Series summation involves adding a sequence of numbers, and when dealing with infinite geometric series, we are adding an unending sequence of numbers that follow a specific pattern. For example, with the repeating decimal 0.013013013..., we can rewrite it as an infinite series: 0.013 + 0.000013 + 0.000000013 + ...
Each term in this series decreases by a factor of 0.001, making this a geometric series where each number is the result of multiplying the previous one by the common ratio, here 0.001. To find the sum of this infinite series, we use a special formula:

• Formula: \(S = \frac{a}{1 - r}\)
• \(a\) is the first term, \(r\) is the common ratio.

Using this formula, we effectively calculate the sum of all the terms in the infinite series.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, such as \(\frac{13}{999}\). These numbers result from the division of integers and can be either terminating or repeating decimals with repeating decimals falling into the rational category.
The importance of representing a repeating decimal as a rational number lies in the simplicity of fractions in mathematical operations. By converting decimals to rational numbers, it becomes easier to perform arithmetic operations and make accurate comparisons. Thus, repeating decimals like 0.013013013... can be transformed into manageable fractions like \(\frac{13}{999}\), making them more tangible and manipulable in various mathematical contexts.

Converting Decimals to Fractions

Converting decimals to fractions involves finding equivalent values that express the same quantity but in a rational form. This is particularly useful for repeating decimals, where a pattern allows us to redefine them as fractions. For instance, with the decimal 0.013013013... :

• Recognize the repeating block: "013".
• Express the repeating decimal as an infinite series, leading to the equation \(S = \frac{a}{1 - r}\).
• Calculate the sum, which gives the fractional form.

The conversion technique highlights the beauty of mathematics in providing precise answers. The calculated equivalent, \(\frac{13}{999}\), is a perfect representation of the repeating decimal, ensuring no loss of precision in calculations.