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 and \(\lim _{n \rightarrow \infty} b_{n}=0\), then \(\sum_{n=1}^{\infty}\left(b_{2 n-1}-b_{2 n}\right)\) converges absolutely.

Short answer

The statement is false; the series does not converge absolutely.

Step-by-step solution

Understanding the Statement

The given sequence \( b_n \) is defined as a sequence of non-negative terms \( b_{n} \geq 0 \), and is decreasing. Additionally, \( \lim_{n \to \infty} b_n = 0 \). We need to determine if the series \( \sum_{n=1}^{\infty} (b_{2n-1} - b_{2n}) \) converges absolutely.

Setting Up the Series

The series \( \sum_{n=1}^{\infty} (b_{2n-1} - b_{2n}) \) is composed of the differences between consecutive terms in the sequence, where each term represents the difference between an odd-ranked and the immediately following even-ranked term.

Analyzing Absolute Convergence

To check for absolute convergence of \( \sum_{n=1}^{\infty} |b_{2n-1} - b_{2n}| \), we consider that since \( b_n \) is decreasing, each difference \( b_{2n-1} - b_{2n} \geq 0 \). Thus \( |b_{2n-1} - b_{2n}| = b_{2n-1} - b_{2n} \). For absolute convergence, \( \sum_{n=1}^{\infty} (b_{2n-1} - b_{2n}) < \infty \) needs to hold.

Counterexample with Harmonic Series

Consider \( b_n = \frac{1}{n} \). This sequence is decreasing and tends to zero, but the series involves terms of the alternating harmonic series pattern. Specifically, we examine \( b_{2n-1} - b_{2n} = \frac{1}{2n-1} - \frac{1}{2n}\), which does not converge absolutely because these differences loosely resemble an alternating series where absolute terms sum toward infinity.

Determining the Final Verdict

Despite individual terms approaching zero, the absolute series \( \sum_{n=1}^{\infty} |b_{2n-1} - b_{2n}| = \sum_{n=1}^{\infty} \left|\frac{1}{2n-1} - \frac{1}{2n}\right| \) essentially rebuilds a harmonic-like pattern. Hence, it does not converge absolutely for the harmonic example given and the statement is false.

Key concepts

Absolute Convergence

Absolute convergence of a series is a stronger type of convergence than conditional convergence. For a series \( \sum_{n=1}^{\infty} a_n \), absolute convergence means that the series of absolute values \( \sum_{n=1}^{\infty} |a_n| \) is convergent.
If a series converges absolutely, it also converges conditionally. However, the reverse is not true.
When a series converges absolutely, the order of terms can be rearranged without affecting the sum. This makes absolute convergence a very robust condition for series convergence.

• To determine if a series converges absolutely, evaluate \( \sum_{n=1}^{\infty} |a_n| \).
• If this series converges, then the original series \( \sum_{n=1}^{\infty} a_n \) also converges absolutely.
• If it does not converge, the original series is not absolutely convergent, though it may converge conditionally.

In the case of the original exercise, absolute convergence is disproven using the harmonic series as a counterexample which does not converge absolutely.

Alternating Series

An alternating series has terms that alternate in sign, exemplified by sequences like \( -1, 1, -1, 1, \ldots \). These series can converge even when individual terms do not decrease rapidly. A typical form of an alternating series is \( \sum_{n=1}^{\infty} (-1)^n a_n \) where \( a_n > 0 \).
This convergence relies on two conditions:

• The sequence \( a_n \) should be eventually decreasing.
• The limit \( \lim_{n \to \infty} a_n = 0 \).

The Alternating Series Test states that such alternating series converge conditionally, as they do not meet the absolute convergence criteria. The original exercise involves a pattern resembling the terms of an alternating series without glazing through absolute convergence.

Harmonic Series

The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is an important example in sequence and series study. It is known to diverge because its terms get smaller very slowly; thus, their infinite sum grows without bound.
An alternating version, such as \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} \), known as the alternating harmonic series, converges conditionally by the Alternating Series Test.

• The non-alternating harmonic series is significant in showing how seemingly small terms can lead to divergence.
• The exercise showcases this by using harmonic-like series terms to explain the lack of absolute convergence.

This underscores why reductions between terms may not suffice to ensure convergence, due to similar properties the harmonic series possesses.

Sequence Behavior

In understanding series convergence, examining how sequence terms \( b_n \) behave is crucial. A sequence is a list of numbers in a specific order, and "behavior" describes how these numbers progress, for instance, increasing, decreasing, or approaching a limit.
For convergence of series \( \sum_{n=1}^{\infty} a_n \), key sequence behaviors include:

• If \( b_n \) is decreasing and \( \lim_{n \to \infty} b_n = 0 \), it suggests potential for convergence, mainly for alternating series.
• Even if terms decrease and become negligible, this alone does not ensure absolute convergence, as highlighted by the counterexample in the original exercise.
• The exercise reveals how careful analysis of sequence term differences, like \( b_{2n-1} - b_{2n} \), is necessary to understand convergence characteristics.

Understanding these behaviors alongside series tests offers deeper insights into the intricacies of convergence. This allows better decision-making when evaluating whether a series converges or not.