Problem 55 - A Graphical Approach to College Algebra

Question

Find each sum that converges. $$\frac{4}{3}+\frac{2}{3}+\frac{1}{3}+\dots$$

Step-by-Step Solution

Answer

The sum of the series is \(\frac{8}{3}\).

Step 1: Identify the Series Type

The given series is \(\frac{4}{3} + \frac{2}{3} + \frac{1}{3} + \dots\). This series is a geometric series where each term is multiplied by a common ratio to get the next term.

Step 2: Find the First Term and the Common Ratio

The first term \(a\) of the series is \(\frac{4}{3}\). To find the common ratio \(r\), we divide the second term by the first term: \(\frac{2}{3} / \frac{4}{3} = \frac{1}{2}\). Thus, \(r = \frac{1}{2}\).

Step 3: Determine if the Series Converges

A geometric series converges if the absolute value of the common ratio \(|r| < 1\). In this series, \(|\frac{1}{2}| = \frac{1}{2} < 1\), so it converges.

Step 4: Use the Geometric Series Sum Formula

The sum \(S\) of an infinite geometric series is given by the formula \(S = \frac{a}{1 - r}\), where \(a\) is the first term and \(r\) is the common ratio.

Step 5: Calculate the Sum

Substitute \(a = \frac{4}{3}\) and \(r = \frac{1}{2}\) into the formula: \[ S = \frac{\frac{4}{3}}{1 - \frac{1}{2}} = \frac{\frac{4}{3}}{\frac{1}{2}} = \frac{4}{3} \times \frac{2}{1} = \frac{8}{3}. \] Thus, the sum of the series is \(\frac{8}{3}\).

Key Concepts

Understanding Series Convergence

A geometric series can either converge or diverge. Convergence of a series means that as you add more and more terms, the total sum approaches a specific, finite value. This is a crucial concept to grasp when dealing with sequences and series. For a geometric series to converge, its common ratio \(r\) must satisfy the condition \( |r| < 1 \).
This condition ensures that each subsequent term becomes smaller and smaller, eventually getting close to zero. Hence, the total sum stabilizes to a finite number. In our example series, \(\frac{4}{3} + \frac{2}{3} + \frac{1}{3} + \dots\), the common ratio \(r\) is \(\frac{1}{2}\). Since \(\left|\frac{1}{2}\right| = \frac{1}{2} < 1\), the series converges.

• Convergence indicates reaching a finite sum.
• If \(|r|\) is greater than or equal to 1, the series diverges.

This means it won't settle down to a specific value as you add more terms.

Calculating the Sum of a Series

Once we determine that a geometric series converges, we can calculate its sum using a specific formula. This formula is a powerful tool for quickly finding this sum without adding an infinite number of terms one by one.
The formula for the sum \(S\) of an infinite geometric series is:
\[ S = \frac{a}{1 - r} \]
Where:

• \(a\) is the first term of the series.
• \(r\) is the common ratio between consecutive terms.

For our example, \(a = \frac{4}{3}\) and \(r = \frac{1}{2}\).
Substituting into the formula gives:
\[ S = \frac{\frac{4}{3}}{1 - \frac{1}{2}} = \frac{\frac{4}{3}}{\frac{1}{2}} = \frac{4}{3} \times 2 = \frac{8}{3} \]
This shows that the sum of the series is \(\frac{8}{3}\). Calculating this sum saves time and is often necessary for understanding properties of geometric series.

Role of the Common Ratio in Series

The common ratio plays a pivotal role in the behavior of a geometric series. It is the factor by which you multiply one term to get the next term in the sequence. In a geometric series, identifying the common ratio is key to understanding whether a series converges or how quickly its terms decrease.
For a series with terms like \(a, ar, ar^2, ar^3, \dots\), each term is gained by multiplying the previous term by \(r\).

• If \(|r| < 1\), terms get progressively smaller, leading to a convergence.
• If \(|r| > 1\), terms grow larger, resulting in divergence.

Our series \(\frac{4}{3} + \frac{2}{3} + \frac{1}{3} + \dots\) has a common ratio \(r = \frac{1}{2}\), which effectively halves each term compared to the one before it.
This consistent reduction in term size ensures convergence and makes calculations manageable. Understanding the common ratio is critical to mastering geometric series and their properties.