Problem 15 - 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). $$ 0.22222 \ldots $$

Step-by-Step Solution

Answer

The decimal 0.22222... as a fraction is \(\frac{2}{9}\).

Step 1: Identify the Decimal Pattern

Observe the decimal given: \(0.22222\ldots\). Notice that it is repeating, with '2' as its repeating digit. This can be written in a generalized form as an infinite series.

Step 2: Express as an Infinite Series

Write the decimal \(0.22222\ldots\) as an infinite series: \[ 0.2 + 0.02 + 0.002 + 0.0002 + ext{etc.} \]. Alternatively, it can be expressed using place values: \[0.2 + 0.2 \times 10^{-1} + 0.2 \times 10^{-2} + 0.2 \times 10^{-3} + ext{etc.}\].

Step 3: Recognize the Geometric Series

The series \( 0.2 + 0.02 + 0.002 + ext{etc.} \) is a geometric series where the first term \(a\) is 0.2 (or \( \frac{2}{10}\)) and the common ratio \(r\) is 0.1 (or \( \frac{1}{10}\)).

Step 4: Use the Sum Formula for Geometric Series

The sum \(S\) of an infinite geometric series can be found using the formula \[ S = \frac{a}{1 - r} \]. Here, \(a = 0.2\) and \(r = 0.1\). Substitute these values into the formula: \[ S = \frac{0.2}{1 - 0.1} = \frac{0.2}{0.9} \].

Step 5: Simplify the Fraction

Simplify \( \frac{0.2}{0.9} \) by multiplying both the numerator and the denominator by 10 to eliminate the decimals: \[ \frac{2}{9} \]. Thus, the sum and the decimal \(0.22222\ldots\) can be expressed as the fraction \( \frac{2}{9} \).

Key Concepts

Repeating Decimals

A repeating decimal is a type of decimal number that has a sequence of digits that repeats indefinitely. For example, in the decimal \(0.22222\ldots\), the digit '2' repeats forever. In general, any repeating decimal can be expressed in mathematical notation by identifying and isolating the repeating part.

The repeating patterns might be a single digit, a group of digits, or more complex sequences. Recognizing these patterns can help convert repeating decimals into fractions, which are easier to work with in calculations.

When you're dealing with repeating decimals, remember that they're not endless non-repeating decimals. They have a predictable pattern, making them manageable through conversion into fractions using geometric series. Look for the simplest repeating sequence and write it out to understand how the decimal is structured.

Geometric Series

A geometric series is a series of terms where each term is a constant multiple (called the common ratio) of the previous term. In this case, the infinite series for \(0.22222\ldots\) is \(0.2 + 0.02 + 0.002 + \ldots\).

In a geometric series, the key components are:

• The first term \(a\): This is the beginning term of the series, which is \(0.2\) in our example (or \(\frac{2}{10}\)).
• The common ratio \(r\): This indicates how each term relates to the next. For \(0.22222\ldots\), the ratio is \(0.1\) (or \(\frac{1}{10}\)). Each term is the previous term multiplied by \(0.1\).

The knowledge of a geometric series helps in calculating its sum, especially when it goes on endlessly.

Sum of Series

The sum of an infinite geometric series can be calculated when the common ratio \(r\) (where \(0 < r < 1\)) is known. The sum formula \(S = \frac{a}{1 - r}\) is essential here, where \(a\) is the first term and \(r\) is the common ratio.

Using our example, for \(0.22222\ldots\), the first term \(a = 0.2\) and the common ratio \(r = 0.1\). Plugging into the formula gives:
\[S = \frac{0.2}{1 - 0.1} = \frac{0.2}{0.9} = \frac{2}{9}\]
This calculation converts the repeating decimal into a simple fraction, making it easier to use in further mathematical operations. The sum of the series provides an exact value, rather than an approximation that decimals often give.