Problem 54 - College Algebra with Corequisite Support

Question

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

Step-by-Step Solution

Answer

The fraction is \(\frac{65}{99}\).

Step 1: Identify the Recurring Decimal

Recognize that the repeating decimal \(0.\overline{65}\) can be written as \(0.656565...\). The repeating block is \(65\).

Step 2: Express as an Infinite Series

Write the decimal as an infinite series: \(0.\overline{65} = 0.65 + 0.0065 + 0.000065 + \ldots\). Each term can be seen as \(65 \times 10^{-2}, 65 \times 10^{-4}, 65 \times 10^{-6}, \ldots\).

Step 3: Define the First Term and Common Ratio

The first term \(a\) of the series is \(0.65\) or \(\frac{65}{100}\). The common ratio \(r\) is \(0.01\), as each term is formed by multiplying the previous one by \(0.01\).

Step 4: Write in Summation Notation

Use summation notation to express the series: \(\sum_{n=0}^{\infty} 65 \times 10^{-(2 + 2n)}\) or equivalently \(\sum_{n=0}^{\infty} \frac{65}{100} \times (0.01)^n\).

Step 5: Apply the Infinite Geometric Series Formula

For an infinite geometric series with first term \(a = \frac{65}{100}\) and common ratio \(r = 0.01\), the sum is \(S = \frac{a}{1-r}\).

Step 6: Calculate the Sum

Substitute \(a = \frac{65}{100}\) and \(r = 0.01\) into the formula: \[S = \frac{\frac{65}{100}}{1-0.01} = \frac{\frac{65}{100}}{0.99} = \frac{65}{99}\]. Simplify if needed, but \(\frac{65}{99}\) is already in simplest form.

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms that follow a specific pattern. When dealing with geometric series, these terms can often be expressed as powers of a common ratio. For example, in an infinite geometric series, each term can be obtained by multiplying the previous term by a certain fixed number known as the common ratio. The concept of an infinite series is essential because it allows us to understand how repeating numbers, like recurring decimals, can be represented as sums going on indefinitely.

In the case of a repeating decimal like \( 0.\overline{65} \), we express it as a never-ending sum: \( 0.65 + 0.0065 + 0.000065 + \ldots \).
This series has a first term (\(a = 0.65\)) and each subsequent term results from multiplying the previous term by the common ratio (\(r = 0.01\)).

Luckily, if a series is geometric and has an absolute common ratio less than 1, we can calculate the sum of this endless series using a simple formula. This is particularly useful for those numerical curiosities that seem hard to catch with traditional arithmetic methods.

Summation Notation

Summation notation is a compact way to represent the sum of several terms in a series. It uses the Greek letter Sigma (\(\Sigma\)) and lets you write an otherwise complex series in a straightforward way. This facilitates understanding and calculations, especially when the number of terms goes on forever, as in infinite series.

For our example, the repeating decimal \(0.\overline{65}\) converts into the series \(0.65 + 0.0065 + 0.000065 + \ldots \). In summation notation, this is written as:

• \(\sum_{n=0}^{\infty} 65 \times 10^{-(2 + 2n)}\)
• Alternatively, \(\sum_{n=0}^{\infty} \frac{65}{100} \times (0.01)^n\)

Here, \(n\) starts at 0 and goes to infinity, each time applying the formula to find the next term in the series. Summation notation elegantly captures the infinite nature of the series, making it a useful tool in mathematics.

Recurring Decimals

A recurring decimal is a decimal number that repeats a finite sequence of digits indefinitely. The repeating part is called a "repetend" and can be written with a bar over it, like \(0.\overline{65}\). Understanding how to convert these decimals into fractions provides deeper insight into number patterns and their exact values.

For example, \(0.\overline{65}\) translates into an infinite geometric series \(0.65 + 0.0065 + 0.000065 + \ldots\). When calculating its sum, it transforms into the fraction \(\frac{65}{99}\).

Here's why this is useful:

• You get a precise value for what seems to be an awkward repeating decimal.
• It aids in calculations where exactness is vital, like financial computations.
• Gaining this math skill helps improve your number sense and familiarity with patterns.

By learning to handle recurring decimals, we peek into how infinite patterns can be entirely defined and represented with simple mathematical expressions.