Problem 44 - A Graphical Approach to College Algebra

Question

Write the sum of each geometric series as a rational number. $$0.36+0.0036+0.000036+\cdots$$

Step-by-Step Solution

Answer

The sum of the series is \( \frac{4}{11} \).

Step 1: Identify the First Term

The first term of the series is given as \( a = 0.36 \). This is the starting point for our geometric series.

Step 2: Determine the Common Ratio

To find the common ratio \( r \), divide the second term by the first term: \( r = \frac{0.0036}{0.36} = 0.01 \).

Step 3: Use the Formula for the Sum of an Infinite Geometric Series

The formula for the sum \( S \) of an infinite geometric series when \(|r| < 1\) is \( S = \frac{a}{1 - r} \).

Step 4: Calculate the Sum

Plug the values of \( a = 0.36 \) and \( r = 0.01 \) into the formula: \( S = \frac{0.36}{1 - 0.01} = \frac{0.36}{0.99} \).

Step 5: Simplify the Fraction

Convert the decimal \( \frac{0.36}{0.99} \) to a simple fraction by multiplying the numerator and denominator by 100: \( \frac{36}{99} = \frac{4}{11} \) after simplification.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Unlike a finite series, which has a definite number of terms and a sum that we can easily find, an infinite series continues indefinitely. The key challenge with infinite series is determining whether they have a sum. Not all infinite series sum to a finite number, but for those that do, there are specific conditions.

• A series converges if it approaches a specific value as more terms are added.
• If a series does not approach a specific value, it diverges.

The geometric series we are focusing on is a type of infinite series which can converge. Convergence in a geometric series depends on the common ratio, which must be between -1 and 1.

Sum of Series

Finding the sum of a series can be simple or complex, depending on the series type. For a geometric series, there's a handy formula that applies only when the series converges. This is the case when the common ratio's absolute value is less than 1.The formula is:\[S = \frac{a}{1 - r}\]where:

• \( a \) is the first term, and
• \( r \) is the common ratio of the series.

This formula helps simplify the process of finding the sum, allowing us to handle complex series with ease. For the given series starting with 0.36 and a common ratio of 0.01, this formula yields a sum of \( \frac{4}{11} \). This simplification shows how an infinite sequence of numbers can add up to a precise rational number.

Common Ratio

The common ratio is crucial in determining the behavior of a geometric series. In essence, it tells us how each term in the series relates to the previous one. A constant factor that you multiply by each time to get to the next term.

• If the common ratio is positive or negative but less than 1 in absolute terms, the series will converge.
• If the ratio is greater than 1 or less than -1, the series will diverge.

For the example at hand, the common ratio is 0.01, meaning each term is 1% of the previous term. By identifying this tiny common ratio, we can easily apply the formula for the sum of an infinite geometric series, helping us quickly resolve and understand the series' total value. Recognizing the role of the common ratio lets you foresee whether to expect a finite sum for infinite series scenarios.