Problem 16 - Thomas Calculus

Question

In Exercises \(15-18,\) determine if the geometric series converges or diverges. If a series converges, find its sum. $$ 1+(-3)+(-3)^{2}+(-3)^{3}+(-3)^{4}+\cdots $$

Step-by-Step Solution

Answer

The series diverges because the common ratio \(|r| = 3 > 1\).

Step 1: Identify the Series Parameters

The given series is a geometric series. Identify the first term \(a\) and the common ratio \(r\). In this series, the first term \(a = 1\) and the common ratio \(r = -3\).

Step 2: Determine if the Series Converges or Diverges

A geometric series converges if the absolute value of the common ratio \(|r| < 1\). In this series, \(|r| = |-3| = 3\). Since \(3 > 1\), the series diverges.

Step 3: Conclude the Series Analysis

Since \(|r| > 1\), the series does not converge. Thus, there is no sum for this series as it diverges.

Key Concepts

Convergence and Divergence

Understanding whether a series converges or diverges is crucial in the study of infinite series. When we talk about convergence or divergence, we are looking at the nature of a series as it extends towards infinity. These terms help us determine if adding up the terms of the series will approach a specific finite number (convergence) or if it will keep growing without bound (divergence).

In a geometric series, convergence depends on the common ratio, which we'll discuss next. A vital rule to remember is that a geometric series converges when the absolute value of its common ratio is less than one,

• A series converges: - If the sum of its terms reaches a specific number over time. - This means that the series approaches a fixed value as more terms are added.
• A series diverges: - If the terms do not sum up to a finite number. - The series might keep increasing or decreasing indefinitely.

By checking \(|r| < 1\) for convergence, we determine if an infinite series will settle at a finite sum, or if the sum grows indefinitely, hence diverging.

Common Ratio

The common ratio in a geometric series plays a pivotal role in determining both the behavior of the series and whether it converges or diverges. In our exercise, the common ratio is found between consecutive terms.

Consider each term in the series as being multiplied by the same number (the common ratio) to get the next term. For example, in the series provided \(1, -3, 9, -27, \ldots \), the common ratio \(r\) is \(-3\). This means that every term is obtained by multiplying the previous term by \(-3\).

• To calculate the common ratio, use the formula to pick any two consecutive terms: \[ r = \frac{{a_{n+1}}}{{a_n}} \] where \(a_{n+1} \) will be the next term and \(a_n \) is the current term.

Knowing the common ratio helps in assessing convergence:

• If \(|r| < 1\) , the series converges.
• If \(|r| \geq 1\) , the series diverges.

Thus, the common ratio helps us predict if we can sum the series up to a specific value or if it'll diverge.

Infinite Series

An infinite series is a sum of an endless sequence of terms. In a geometric infinite series, each term is determined by multiplying the previous term by a constant called the common ratio. If the series converges, we can find its sum.

For a series to be considered an infinite geometric series, it must continue indefinitely. The terms never stop, and theoretically, the series stretches to infinity.

• Geometric Series: - Defined by a starting term and a constant ratio. - In our case, the first term \(a\) is \(1\), and the common ratio \(r\) is \(-3\).

However, an infinite series can only have a finite sum if it converges. The sum of a convergent geometric series is calculated using the formula:\[S = \frac{a}{1 - r}, \text{ for } |r| < 1\]This formula is not applicable if \(|r| \geq 1\) because the series is divergent, as in our given series, where \(|-3| = 3\), which is greater than 1, hence the series diverges. Understanding infinite series helps identify their different behaviors and whether they have a finite sum or not.