Problem 17 - Calculus: Early Transcendentals

Question

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \( 3 - 4 + \frac {16}{3} - {64}{9} + \cdot \cdot \cdot \)

Step-by-Step Solution

Answer

The series is divergent.

Step 1: Identify the First Term and Common Ratio

In a geometric series, the first term is denoted by \( a \). Here, the first term \( a = 3 \).To find the common ratio \( r \), divide the second term by the first term: \(-4 / 3 = -\frac{4}{3}\). Therefore, \( r = -\frac{4}{3} \).

Step 2: Determine Convergence or Divergence

A geometric series converges if the absolute value of the common ratio \( |r| < 1 \). Here, the common ratio \( r = -\frac{4}{3} \), so \(|r| = \frac{4}{3} \).Since \(|r| > 1\), the series is divergent.

Key Concepts

Convergent Series

Understanding when a geometric series converges is crucial for dealing with infinite sums. A series is said to be convergent if its terms approach a specific finite limit as they are summed infinitely. In simpler terms, if we keep adding more terms from the series, a convergent series will settle to a particular value rather than increasing indefinitely or fluctuating wildly.
To determine if a geometric series converges, we examine its common ratio, denoted as \( r \). For a geometric series with an infinite number of terms, it converges only if the absolute value of its common ratio \(|r|\) is less than 1 (\(|r| < 1\)). When this condition is met, the series will converge to a certain sum, and this sum can actually be calculated using the formula:

• Sum = \( \frac{a}{1 - r} \)

where \( a \) is the first term of the series.
If this condition is not satisfied, the series doesn't converge, and it's called a divergent series.

Divergent Series

A geometric series is considered divergent if it doesn't settle down to a particular value as more terms are added. This happens when the absolute value of the common ratio \(|r|\) is greater than or equal to 1. In such cases, the terms of the series keep growing without bound or oscillate, making it impossible to define a finite sum for the series.
In the context of the provided exercise, the series:

• 3, -4, \( \frac{16}{3} \), -\( \frac{64}{9} \) ...

turns out to be divergent because the common ratio \( r = -\frac{4}{3} \) has an absolute value greater than 1. Therefore, the series cannot possibly converge to a simple number. Instead, as you add more and more terms, the total sum keeps changing without settling down.
This kind of understanding is crucial for working with series in calculus and higher mathematics.

Common Ratio

The common ratio is a key concept in understanding geometric series. It is what sets the pattern for the series, determining the relationship between consecutive terms.
To find the common ratio \( r \) in a geometric series, you simply divide any term in the series by the preceding one. For example, in a series where the first few terms are given, you take the second term and divide it by the first:

• \( r = \frac{\text{second term}}{\text{first term}} \)
• In the exercise: \( -4 / 3 = -\frac{4}{3} \)

This value, \( -\frac{4}{3} \), is the common ratio.
The common ratio ultimately determines whether the series will converge or diverge, depending on its absolute value. Understanding the common ratio helps in predicting the long-term behavior of the series, which is central to solving problems involving geometric progressions.

First Term

The first term of a geometric series is the starting point and is a crucial element when trying to compute the sum of a series, especially if it converges. The first term is usually denoted by \( a \) and identifies the initial value of the series.
In the exercise problem, the first term \( a \) is clearly given as 3. This term is important not only for understanding the structure of the series but also for calculating convergence sums when applicable.

• If a series were convergent, the first term is used in conjunction with the common ratio \( r \) to find the sum using the formula \( \text{Sum} = \frac{a}{1 - r} \)

Although the series in this exercise is divergent and such a sum can't be calculated, the first term still sets the stage for the sequence and its progression. Being clear about this initial term helps in accurately analyzing and understanding a given geometric series.