Problem 22 - Thomas Calculus

Question

In Exercises \(15-22\) , determine if the geometric series converges or diverges. If a series converges, find its sum. $$\frac{9}{4}-\frac{27}{8}+\frac{81}{16}-\frac{243}{32}+\frac{729}{64}-\cdots$$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the First Term and the Common Ratio

The given series is \( \frac{9}{4} - \frac{27}{8} + \frac{81}{16} - \frac{243}{32} + \frac{729}{64} - \cdots \). The first term of the series \( a \) is \( \frac{9}{4} \). To find the common ratio \( r \), divide the second term by the first term: \(-\frac{27}{8} \div \frac{9}{4} = -\frac{27}{8} \times \frac{4}{9} = -\frac{3}{2}\). Thus, the common ratio \( r \) is \(-\frac{3}{2}\).

Step 2: Determine Convergence or Divergence

To check if the series converges, we must examine the absolute value of the common ratio \( r \). A geometric series converges if \(|r| < 1 \). Here, \(|r| = \left|-\frac{3}{2}\right| = \frac{3}{2} \), which is greater than 1. Therefore, the series diverges.

Key Concepts

Convergence and Divergence

A geometric series can either converge or diverge. To determine this, we examine the common ratio, denoted as \( r \). Convergence means that the series approaches a specific value as the number of terms goes to infinity. On the other hand, divergence indicates that the series grows without bound or does not settle to a particular value. For a geometric series, convergence occurs if the absolute value of the common ratio \( |r| \) is less than 1. This restriction ensures that successive terms become smaller, allowing the sum to stabilize. However, if \(|r| \geq 1\), the series is divergent as the terms either remain the same size or grow larger, preventing the series from settling into a stable sum.

Common Ratio

The common ratio in a geometric sequence or series is a crucial quantity that helps determine the behavior of the series. It is defined as the factor by which we multiply each term to get the next term in the series. Given two consecutive terms, the common ratio \( r \) is found by dividing the second term by the first term. For instance, in the exercise series \( \frac{9}{4}, -\frac{27}{8}, \frac{81}{16}, \cdots \), the common ratio is calculated by dividing \( -\frac{27}{8} \) by \( \frac{9}{4} \), which equals \( -\frac{3}{2} \). This ratio tells us that each term is multiplied by \( -\frac{3}{2} \) to obtain the subsequent term in the sequence.

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio \( r \). In the context of a geometric series, these sequences are summed to form the series. For example, the sequence in our exercise starts with \( \frac{9}{4} \) and continues with subsequent terms like \( -\frac{27}{8}, \frac{81}{16} \), and so forth, following a pattern dictated by the common ratio \( -\frac{3}{2} \). Each term is derived from the previous one by applying the common ratio, illustrating the consistent relationship throughout the series.

Series Sum Formula

When a geometric series converges, there is a straightforward formula for finding the sum of the infinite series. The sum \( S \) of an infinite geometric series is given by \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. This formula applies only when the series converges, i.e., when \(|r| < 1\). However, in the case of the original exercise, where \( r = -\frac{3}{2} \), the series diverges because \( |r| \) is greater than 1. Thus, the series does not have a finite sum. The formula emphasizes the importance of the common ratio in identifying whether the sum of the series is finite.