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 \(b_{n} \geq 0\) is decreasing, then \(\sum_{n=1}^{\infty}\left(b_{2 n-1}-b_{2 n}\right)\) converges absolutely.

Short answer

False. Consider \( b_n = \frac{1}{n} \); the series does not converge absolutely.

Step-by-step solution

Understand the Problem

We need to determine if the series \( \sum_{n=1}^{\infty}(b_{2n-1}-b_{2n}) \) converges absolutely given that \( b_n \geq 0 \) is decreasing.

Analyze the Series

Consider the terms of the series \( (b_{2n-1} - b_{2n}) \). If \( b_n \) is decreasing, then \( b_{2n-1} \geq b_{2n} \), making each term non-negative.

Consider Absolute Convergence

A series converges absolutely if \( \sum_{n=1}^{\infty} |b_{2n-1} - b_{2n}| \) converges. Since each term \( |b_{2n-1} - b_{2n}| = b_{2n-1} - b_{2n} \) is already non-negative, we actually check if \( \sum_{n=1}^{\infty} (b_{2n-1} - b_{2n}) \) converges.

Construct a Counterexample

Let's choose \( b_n = \frac{1}{n} \), which is positive and decreases. Now consider the series terms: \( b_{2n-1} = \frac{1}{2n-1} \) and \( b_{2n} = \frac{1}{2n} \). The term \( b_{2n-1} - b_{2n} = \frac{1}{2n-1} - \frac{1}{2n} \).

Simplify the Counterexample

Simplifying, we have \( b_{2n-1} - b_{2n} = \frac{2n - (2n-1)}{(2n-1)(2n)} = \frac{1}{(2n-1)(2n)} \). The series \( \sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n)} \) resembles the Harmonic series which diverges, indicating \( b_{2n-1} - b_{2n} \) does not converge absolutely.

Conclusion

Since for \( b_n = \frac{1}{n} \), the series \( \sum_{n=1}^{\infty} (b_{2n-1} - b_{2n}) \) does not converge absolutely, the statement is false.

Key concepts

Absolute Convergence

When we talk about absolute convergence, we mean that the series will still converge even if we take the absolute value of each term. If a series \( \sum a_n \) converges absolutely, it means that \( \sum |a_n| \) also converges. This is a stronger form of convergence compared to simple or conditional convergence. It ensures stability in convergence since taking the absolute value usually increases the sum's magnitude.

• Absolute convergence means \( \sum |a_n| \) is finite.
• If a series converges absolutely, then it converges in the normal sense.
• It implies that the series' terms collectively move closer to a limit with their variations smoothed out.

To determine absolute convergence, switch out the original values in your series with their absolute values and check if the series still converges. If not, then the series does not converge absolutely.

Harmonic Series

The harmonic series is the sum \( \sum_{n=1}^{\infty} \frac{1}{n} \). It is a classic example of a divergent series, even though its individual terms tend to zero. This often surprises students because the terms become very small. However, their rate of decrease isn't fast enough for convergence.

• The harmonic series diverges: \( \sum_{n=1}^{\infty} \frac{1}{n} = \infty \).
• Even splitting it into subsets, like half-terms or every other term, tends to maintain divergence.
• Comparisons with other series often show why the harmonic series diverges when it seems it should converge.

Understanding its divergence helps in analyzing more complex series by comparison or by recognizing subsets or similar structures within those series.

Decreasing Sequence

A decreasing sequence is a sequence where each term is less than or equal to the preceding one. Mathematically, a sequence \( b_n \) is decreasing if \( b_n \geq b_{n+1} \) for all \( n \).

• A decreasing sequence is useful for testing series convergence.
• Even if a sequence decreases, it doesn't necessarily ensure the corresponding series will converge.
• Taken together with the positivity condition \( b_n \geq 0 \), a decreasing sequence often helps form test examples and counterexamples.

In mathematical problems, being given a decreasing sequence often aids in simplifying or reconceptualizing series and helps understand the behavior of the terms within it.

Counterexample in Mathematics

Counterexamples play a crucial role in mathematics. They help demonstrate that a statement is false by providing a specific case where the statement doesn't hold. In series and sequences, a counterexample can reveal the boundaries of where a given rule or pattern might work.

• It confirms the limitations or exceptions of mathematical statements.
• Counterexamples are single examples for which the statement does not satisfy the given conditions.
• In our problem, taking \( b_n = \frac{1}{n} \) served as a counterexample to show that the series does not converge absolutely.

Through careful construction and analysis of counterexamples, mathematicians gain a deeper understanding of principles and their exceptions, paving the way for a more profound grasp of mathematical theories and their applications.