Problem 71 - Algebra for College Students

Question

What do we mean when we say that the infinite geometric sequence \(1,2,4,8, \ldots\) has no sum?

Step-by-Step Solution

Answer

The sequence diverges because the common ratio is greater than 1.

Step 1: Understanding the Problem

The infinite geometric sequence given is \(1, 2, 4, 8, \ldots\). This means that each term after the first term is obtained by multiplying the previous term by 2. We need to determine why this sequence has no finite sum.

Step 2: Identifying the Sequence Formula

The general formula for a geometric sequence is \(a, ar, ar^2, ar^3, \ldots\) where \(a\) is the first term and \(r\) is the common ratio. For the sequence \(1, 2, 4, 8, \ldots\), \(a = 1\) and \(r = 2\).

Step 3: Evaluating the Common Ratio

For a geometric series to have a sum, the common ratio \(r\) must satisfy \(|r| < 1\). In this sequence, \(r = 2\) which means \(|r| = 2\). Since \(2 > 1\), the condition for convergence is not met.

Step 4: Conclusion: Divergence of Series

Since the common ratio \(r = 2\) is greater than 1, each term in the sequence increases exponentially, and hence the series diverges. An infinite geometric sequence with \(|r| \geq 1\) does not have a finite sum.

Key Concepts

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. When we add up all terms in a geometric sequence, we get what is called a geometric series. In a simple sequence like \(1, 2, 4, 8, \ldots\), every term is obtained by multiplying the previous one by a fixed number, known as the common ratio. Here, we are dealing with the sum \(S = 1 + 2 + 4 + 8 + \ldots\). The series can continue infinitely, leading us to the concept of an infinite geometric series.

• If the sum converges, we can find a finite answer.
• If it diverges, the total sum simply grows larger without limit.

To check for convergence, we need to examine the common ratio, a topic we’ll explore more closely.

Convergence and Divergence

Convergence and divergence are key ideas in determining whether a geometric series has a finite sum. **Convergence** refers to the scenario where the sum of the infinite series approaches a finite number. For a geometric series to converge, the absolute value of the common ratio \(|r|\) must be less than 1. This ensures the terms get progressively smaller, nearing zero, and allowing the series to sum up to a finite number.
On the other hand, **divergence** happens when \(|r|\) is equal to or greater than 1. In these cases:

• The terms do not get smaller, meaning they do not tend to zero.
• The sum of the series grows indefinitely.

In the sequence \(1, 2, 4, 8, \ldots\), the common ratio is 2 (\(|r| = 2\)), which causes the series to diverge because the sum keeps increasing indefinitely.

Common Ratio

The common ratio \(r\) is an essential element in understanding geometric sequences and series. It is calculated by dividing any term in the sequence by the previous term and remains constant throughout the sequence. For the sequence \(1, 2, 4, 8, \ldots\), the common ratio is \(2\) as each term is double the preceding one.
The common ratio determines the behavior of the series regarding convergence and divergence.

• If \(|r| < 1\), the terms become progressively smaller, leading to convergence.
• If \(|r| \geq 1\), the terms increase or stay the same, causing divergence.

Understanding the common ratio helps in predicting the nature and sum of the entire series.