Question

For the following exercises, indicate whether each of the following statements is true or false. If the statement is false, provide an example in which it is false.If \(b_{n} \geq 0\) and \(\lim _{n \rightarrow \infty} b_{n}=0\) then \(\sum_{n=1}^{\infty}\left(\frac{1}{2}\left(b_{3 n-2}+b_{3 n-1}\right)-b_{3 n}\right)\) converges.

Short answer

False. The series diverges if \(b_n = \frac{1}{n}\).

Step-by-step solution

Analyze the Series

Given the series \[ \sum_{n=1}^{\infty}\left(\frac{1}{2}\left(b_{3 n-2}+b_{3 n-1}\right)-b_{3 n}\right) \]We need to determine whether this series converges by analyzing its terms.

Use Limit Comparison

For the series to converge, the terms should approach zero as \(n \rightarrow \infty\). Each term of the series is \[ \frac{1}{2}(b_{3n-2} + b_{3n-1}) - b_{3n} \]Using the limit property \(\lim_{n \rightarrow \infty} b_{n} = 0\),we find that:1. \(b_{3n-2} \rightarrow 0\), 2. \(b_{3n-1} \rightarrow 0\), and 3. \(b_{3n} \rightarrow 0\).Thus, the terms themselves approach \( \frac{1}{2}(0 + 0) - 0 = 0 \).

Construct an Example

To see whether the series converges, consider that the alternating elements could cancel or sum to a positive number. Take \(b_n = \frac{1}{n}\); then,\[ b_{3n-2} = \frac{1}{3n-2}, \quad b_{3n-1} = \frac{1}{3n-1}, \quad \text{and} \quad b_{3n} = \frac{1}{3n} \]This makes the series terms: \[ \frac{1}{2}\left( \frac{1}{3n-2} + \frac{1}{3n-1} \right) - \frac{1}{3n} \] certainly not convergent as it reduces to positive terms similar to the harmonic series \( \sum \frac{1}{n} \).

Conclusion on Convergence

In the example provided where \(b_n = \frac{1}{n}\), the terms do not converge to zero quickly enough to result in an absolutely convergent series. Since this counterexample demonstrates divergence, the original statement is false.

Key concepts

Series Analysis

Series analysis involves understanding the behavior of series and evaluating whether they converge or diverge. To determine convergence or divergence, we examine the terms of the series and use various tests and methods. A series is generally expressed as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) is the sequence of terms that we are summing over an infinite range of \( n \). If the partial sums of the series, \( S_N = a_1 + a_2 + ... + a_N \), approach a specific constant value as \( N \rightarrow \infty \), the series converges. Otherwise, it diverges.

• A convergent series has a finite limit.
• A divergent series does not have a finite limit.

Understanding whether a series converges is crucial, as it can impact mathematical models and computations where inputs are assumed to be stable (i.e., convergent). Employing series analysis effectively can help identify and correct assumptions or models that don't hold in infinite or real-world scenarios.

Limit Comparison

The limit comparison test is a powerful tool in determining the convergence of series. This test compares a given series to a second, known series to infer whether it behaves the same way in terms of convergence. To use the limit comparison test, follow these steps:
1. Identify the series \( \sum a_n \) that you want to test for convergence.2. Choose a second series \( \sum b_n \) with known convergence properties.3. Calculate the limit \( \lim_{n \rightarrow \infty} \frac{a_n}{b_n} \).

• If this limit is a positive finite number, then both series either converge or diverge together.
• If the limit is zero and the series \( \sum b_n \) converges, then \( \sum a_n \) also converges.
• If the limit is infinite and the series \( \sum b_n \) diverges, then \( \sum a_n \) also diverges.

The limit comparison test is valuable because it simplifies the otherwise complex analysis by reducing it to comparison with a simpler series, typically a well-known benchmark series.

Counterexample in Series Convergence

Sometimes, proving a mathematical statement false requires just a single counterexample. When a claim is made about series convergence, we might use a specific series as a counterexample to show that the claim does not hold universally. A counterexample in series convergence typically involves showing that certain assumptions or conditions do not lead to the claimed result.
For instance, if we have a statement such as "If \( b_n \geq 0 \) and \( \lim_{n \rightarrow \infty} b_n = 0 \), then the series \( \sum_{n=1}^{\infty} a_n \) converges," one might construct an example like picking \( b_n = \frac{1}{n} \). The partial sums of this series, which resemble the harmonic series, diverge. Here, even though \( b_n \) approaches zero as \( n \) increases, it doesn't do so quickly enough to guarantee convergence.

• Counterexamples are crucial for testing the boundaries of mathematical theory.
• They help refine existing models and understandings by proving exceptions.
• Finding and understanding counterexamples enhances deeper mathematical insight.

Therefore, counterexamples play a key role in analyzing and understanding the limitations of mathematical statements related to series convergence.