Problem 53 - College Algebra with Modeling and Visualization

Question

Write each national number in the form of an infinite geometric series. $$ \frac{9}{11} $$

Step-by-Step Solution

Answer

\( \frac{9}{11} = 0.81\overline{81} = 81 \cdot 10^{-2} + 81 \cdot 10^{-4} + 81 \cdot 10^{-6} + \ldots \).

Step 1: Express the Number as a Repeating Decimal

The national number \( \frac{9}{11} \) can be expressed as a repeating decimal. By performing long division, we find that \( \frac{9}{11} = 0.81\overline{81} \), where \( 81 \) repeats indefinitely.

Step 2: Identify the Repeating Block for the Infinite Series

In the decimal \( 0.81\overline{81} \), the repeating block is \( 81 \), which we can use to express the number as a series. Recognize that this is the part we'll recursively build our series upon.

Step 3: Construct the Infinite Geometric Series

Each cycle of the repeating block \( 0.81 \) can be broken down to \( 81 \times 10^{-2} = 0.81 \). Thus, the number can be expressed as an infinite series:\[0.81 + 0.0081 + 0.000081 + \ldots\]This is equivalent to:\[81 \cdot 10^{-2} + 81 \cdot 10^{-4} + 81 \cdot 10^{-6} + \ldots\]

Step 4: Write the Series Using the Formula for an Infinite Geometric Series

This series is geometric with the first term \( a = 81 \times 10^{-2} \) and common ratio \( r = 10^{-2} \). The formula for the sum of an infinite geometric series is \( \frac{a}{1 - r} \):\[\frac{81 \times 10^{-2}}{1 - 10^{-2}} = \frac{81 \cdot 0.01}{0.99} = \frac{9}{11}\]This confirms that the series expression is equivalent to the original fraction.

Key Concepts

Repeating Decimals

When we talk about repeating decimals, we refer to decimal numbers where one or more digits repeat indefinitely. These are often represented with a line over the repeating section. For example, in the repeating decimal \( 0.81\overline{81} \), the digits "81" repeat forever.

Repeating decimals are common when you convert fractions into decimals, especially when the denominator is not a factor of 10. This reflects the cyclic nature of certain divisions. In our example with \( \frac{9}{11} \), performing long division shows how these two digits keep cycling after the decimal point. Understanding this concept is crucial as it allows us to represent the decimal as an infinite series, showcasing how math beautifully handles infinity.

Geometric Series Formula

The geometric series formula is a mathematical way of finding the sum of a series where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. In the case of our repeating decimal \( 0.81\overline{81} \), we can break it down to form an infinite geometric series. This means:

• The series repeatedly adds smaller and smaller segments of the number 81, beginning with 0.81, then 0.0081, then 0.000081, and so on.
• Each segment is a power of 10 smaller than the one before, creating a pattern.
• Using the formula for the sum of an infinite geometric series, \( \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio, we can compute the exact value of such a series.

Through this formula, we connect the repeating decimal back to its original fractional form.

Long Division

Long division is a method that helps us divide two numbers to find the quotient and remainder. It is a powerful tool in handling fractions and converting them into repeating decimals. In the context of the given problem, we applied long division to \( \frac{9}{11} \) to find its decimal form:

• We began by dividing 9 by 11.
• Since 9 is not divisible by 11, we added a decimal point and zero, converting this into 90.
• Continuing the division process reveals the repeating nature of the quotient, here, "81."

Understanding long division is essential, as it lays the groundwork for transforming numbers from fractional to decimal form and helps in identifying repeating patterns.

Common Ratio

In any geometric series, the common ratio is the factor by which we multiply each term to get to the next. It symbolizes the consistent growth or reduction from term to term. For the series derived from \( 0.81\overline{81} \), the common ratio is \( 10^{-2} \) or 0.01.

This value indicates how each subsequent term is 0.01 times the previous term, steadily reducing the size of incremental parts of the series:

• The first term is \( 81 \cdot 10^{-2} \).
• The second term is \( 81 \cdot 10^{-4} \) (which is \( 81 \cdot 10^{-2} \cdot 10^{-2} \)).
• And so forth, each term getting increasingly smaller.

Recognizing this ratio allows us to sum the series accurately through the formula for infinite geometric series and reinforces the interconnectedness between repeating decimals and fractional representations.