Problem 54 - College Algebra

Question

Write \(0 . \overline{65}\) as an infin te geometric series using summation notation. Then use the formula for fi ding the sum of an infin te geometric series to convert 0.65 to a fraction.

Step-by-Step Solution

Answer

The fraction equivalent to \(0.\overline{65}\) is \(\frac{65}{99}\).

Step 1: Identify the Repeating Decimal

The decimal number given is \(0.\overline{65}\), which indicates that 65 is the repeating part.

Step 2: Express as an Infinite Sum

Write \(0.\overline{65}\) as an infinite geometric series: \(0.\overline{65} = 0.65 + 0.0065 + 0.000065 + \ldots\).

Step 3: Find the First Term and Common Ratio

In the geometric series, the first term \(a = 0.65\) and the common ratio \(r = 0.01\) because each subsequent term is a multiplication of the previous one by \(0.01\).

Step 4: Use the Formula for Infinite Geometric Series

The sum of an infinite geometric series is given by \(S = \frac{a}{1 - r}\). Substitute \(a = 0.65\) and \(r = 0.01\) into this formula.

Step 5: Calculate the Sum

Substitute the values: \(S = \frac{0.65}{1 - 0.01} = \frac{0.65}{0.99}\).

Step 6: Convert to a Fraction

Calculate \(\frac{0.65}{0.99}\). Simplify by multiplying the numerator and denominator by 100 to get \(\frac{65}{99}\).

Step 7: Final Answer

So, the fraction form of \(0.\overline{65}\) is \(\frac{65}{99}\).

Key Concepts

Repeating Decimal

A repeating decimal is a decimal number where a digit or a group of digits repeats indefinitely. For example, in the number \(0.\overline{65}\), the digits "65" repeat over and over again.
Rewriting a repeating decimal as a series allows us to express it in mathematical forms that are easier to work with, like fractions or sums. In our exercise, \(0.\overline{65}\) is represented as \(0.65 + 0.0065 + 0.000065 + \ldots\) showing a repeating pattern. This illustrates the transition from a repeating decimal into a series.

Infinite Series

An infinite series is a sum of infinitely many terms. When looking at repeating decimals like \(0.\overline{65}\), they can be expressed as an infinite geometric series.
Using the example of \(0.65 + 0.0065 + 0.000065 + \ldots\), each term represents a smaller contribution adding up to the whole value of the repeating decimal.

• Each new term is formed by multiplying the previous term by a constant, known as the common ratio \(r\).
• In this case, the series follows a pattern where each term is the previous term multiplied by \(0.01\).

The concept of an infinite series helps us appreciate how repeating decimals build up their value through ongoing contributions.

Sum of a Series

Finding the sum of a geometric series, especially an infinite series, allows us to consolidate an endless number of terms into a finite expression.
This is done using the formula for the sum of an infinite geometric series: \[ S = \frac{a}{1-r} \]

• Here, \(a\) is the first term of the series.
• \(r\) is the common ratio, and must satisfy \(|r| < 1\) for the series to converge.

Applying this to \(0.65 + 0.0065 + 0.000065 + \ldots\), we calculate the sum as \(\frac{0.65}{0.99}\). This formula wonderfully simplifies an endless sum into something manageable and meaningful.

Fraction Conversion

Converting the sum of a series from a decimal to a fraction ensures precision and a clean numeric representation.
In our example, we need to convert the sum of \(\frac{0.65}{0.99}\) to a fraction. By multiplying the numerator and denominator by 100, to eliminate the decimal point, we get \(\frac{65}{99}\).

• This step requires simplifying if possible, though in this case, \(\frac{65}{99}\) is already in simplest form.

Understanding fraction conversion from decimal forms helps in diverse aspects of mathematics where exact numbers are crucial, such as algebra and probability.