Problem 20 - 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). $$ 0.36717171 \ldots $$

Step-by-Step Solution

Answer

The decimal 0.36717171... is \(\frac{3581}{9900}\) as a fraction.

Step 1: Understand the Decimal Fraction

The decimal given is 0.36717171... This is a repeating decimal where the part '71' repeats indefinitely after the initial '0.36'. We can express this as 0.36 + 0.001717171... where '71' repeats.

Step 2: Express the Repeating Part as a Series

The repeating part 0.001717171... can be expressed as an infinite geometric series: \(0.0017 + 0.000017 + 0.00000017 + \ldots\), where each term is part of a geometric sequence with the first term \(a = 0.0017\) and common ratio \(r = 0.01\).

Step 3: Find the Sum of the Series

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(r\), where \(|r| < 1\), is given by the formula \(S = \frac{a}{1-r}\). Here, \(a = 0.0017\) and \(r = 0.01\), so \(S = \frac{0.0017}{1-0.01} = \frac{0.0017}{0.99}\). Calculating this gives \(S \approx 0.00171717...\).

Step 4: Sum the Series and Initial Non-Repeating Part

To write the original decimal as a fraction, add the non-repeating part 0.36 to the sum of the repeating series: \(0.36 + 0.00171717...\). The repeating series is \(\frac{0.0017}{0.99}\), so add these: \(0.36 + \frac{0.0017}{0.99}\).

Step 5: Simplify and Convert to a Fraction

To simplify, we need common denominators. \(0.36 = \frac{36}{100}\), and the repeating part has been calculated as \(\frac{0.0017}{0.99} = \frac{17}{9900}\). Convert \(0.36\) as \(\frac{3564}{9900}\). Thus the combined fraction is \(\frac{3564}{9900} + \frac{17}{9900} = \frac{3581}{9900}\).

Step 6: Check and Simplify the Fraction

To ensure the fraction is in its simplest form, find the greatest common divisor (GCD) of 3581 and 9900. Since their GCD is 1, the fraction \(\frac{3581}{9900}\) is already in its simplest form.

Key Concepts

Geometric Series

A geometric series is a fascinating mathematical concept that involves a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. This type of series is easy to identify and understand due to its repetitive multiplicative nature. It starts with an initial term often labeled as \( a \), and then each subsequent term is obtained by multiplying the previous term by the common ratio \( r \).

For example, in the sequence \( a, ar, ar^2, ar^3, \ldots \), we have a geometric series. The terms are formed as consecutive powers of \( r \) multiplied by the first term \( a \).

• First Term, \( a \): The starting point of the series.
• Common Ratio, \( r \): The consistent multiplier applied to each term to get to the next.
• Sum Formula: The infinite series sum can be found using \( S = \frac{a}{1-r} \), provided \( |r| < 1 \).

Understanding these fundamentals equips us to tackle infinite series problems, especially when they intertwine with repeating decimals as shown in many exercises.

Repeating Decimals

Repeating decimals are decimals that have one or more digits after the decimal point that repeat infinitely. These are expressed with a bar over the repeating sequence, for example, \( 0.36717171\ldots \), which can be denoted as \( 0.36\overline{71} \).

In calculations:

• The part to the right of the decimal point before any repetition is the non-repeating component.
• The digits that appear in a cycle after that point are the repeating part.

Repeating decimals often signal the presence of a geometric series once expressed mathematically. Decomposing such numbers into a sum of series makes it easier to convert them into fractions, providing a way to represent these decimals as exact ratios of integers.

Fraction Conversion

Fraction conversion is a crucial step to transform repeating decimals into a more precise numerical representation, using fractions. This process typically involves several steps, such as isolating the repeating part and transforming it into a geometric series.

For instance, to convert \( 0.36\overline{71} \) to a fraction, approach it step-by-step:

• Identify the non-repeating and repeating parts.
• Express the repeating segment as a geometric series.
• Calculate the series sum using the formula \( S = \frac{a}{1-r} \).
• Add the result to the non-repeating part, ensuring a common denominator.

This meticulous approach ensures accuracy and offers a clear path to deducing a fractional equivalent from an infinite decimal sequence.

Sum of Series

The sum of a series, particularly in the infinite context, refers to adding up an endless list of numbers, usually structured through a mathematical pattern like the geometric series. The concept of summing an infinite series initially seems paradoxical—after all, infinite implies never-ending! However, with geometric series, if the absolute value of the common ratio (\( r \)) is less than one, the series does have a finite sum.

By using the formula for the sum \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( |r| < 1 \), mathematicians can calculate a finite sum from infinite additions, enabling decisive and accurate conversions from repeating decimals to fractions.

Such calculations demystify abstract limitless proceedings into definitive values, bridging the gap between continuous expressions and precise arithmetic results.