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 \(\sum_{n=1}^{\infty}(-1)^{n-1} b_{n}\) converges conditionally but not absolutely, then \(b_{n}\) does not tend to zero.

Short answer

The statement is false; a counterexample is \( b_n = \frac{1}{n} \).

Step-by-step solution

Understand the Problem

We are given that the series \( \sum_{n=1}^{\infty} (-1)^{n-1} b_n \) converges conditionally but not absolutely. We need to analyze whether the statement that \( b_n \) does not tend to zero is true or false based on the given information.

Definition of Conditional Convergence

A series \( \sum (-1)^{n-1} b_n \) converges conditionally if it converges, but the series \( \sum b_n \) (without the alternating sign) diverges. This means that \( b_n \) must tend to zero for the alternating series test to hold.

Implication for \(b_n\)

For any alternating series \( \sum (-1)^{n-1} b_n \) to converge, the necessary condition (by the alternating series test) is that \( b_n \to 0 \) as \( n \to \infty \). So, if the series converges, \( b_n \) must tend to zero.

Assess the Statement

The statement claims that if the series converges conditionally, then \( b_n \) does not tend to zero. From the outcome of Step 3, we see that this contradicts the necessary condition for conditional convergence where \( b_n \) should go to zero.

Provide a Counterexample

Suppose \( b_n = \frac{1}{n} \). Then the series \( \sum (-1)^{n-1} \frac{1}{n} \) (alternating harmonic series) converges conditionally, but \( \sum \frac{1}{n} \) (harmonic series) diverges. Here, clearly, \( \frac{1}{n} \to 0 \) as \( n \to \infty \). This directly contradicts the statement and serves as a counterexample.

Key concepts

Alternating Series Test

An alternating series is a series whose terms are alternately positive and negative. The Alternating Series Test helps determine if such a series converges. For the series \( \sum (-1)^{n-1} b_n \), this test states two main conditions for convergence:

• The sequence \( b_n \) must be decreasing. This means that each term is less than or equal to the term before it.
• The term \( b_n \) must tend to zero as \( n \to \infty \).

If both conditions are met, the alternating series converges.
In the original exercise, the series was given to converge conditionally, meaning that while the alternating sequence converges, the corresponding non-alternating series diverges. This confirms that \( b_n \to 0 \) is necessary, despite the misleading original statement.

Convergence and Divergence

Convergence refers to the property of a series whose sum approaches a finite value as the number of terms increases. If a series converges, its terms become smaller and closer to zero over time. Divergence, on the other hand, means the series does not settle into a finite sum but continues growing indefinitely or oscillates without approaching any limit.
In the context of conditional convergence, a series can exhibit both behaviors depending on its form. The alternating series might converge, meeting the test's criteria, while its corresponding non-alternating version diverges. This distinction is crucial for understanding series like the alternating harmonic series, which converge conditionally.

Harmonic Series

The harmonic series is a classic example of a divergent series. Represented by \( \sum \frac{1}{n} \), it is the sum of the reciprocals of the natural numbers:
\[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \]Despite the terms decreasing and tending towards zero, the harmonic series grows without bound.
Interestingly, an alternating version of this series, known as the alternating harmonic series, does converge to a finite value. This illustrates the phenomenon where an alternating sequence can achieve conditional convergence even when its non-alternating counterpart diverges.

Counterexample

Counterexamples play a vital role in mathematics by demonstrating the falsehood of a proposition. In the original exercise, the statement suggested that the sequence \( b_n \) does not tend to zero when the series converges conditionally. However, a specific example can disprove this claim.
Consider \( b_n = \frac{1}{n} \), which forms the basis of the harmonic series. The corresponding alternating series \( \sum (-1)^{n-1} \frac{1}{n} \), known as the alternating harmonic series, actually converges conditionally. Here, \( \frac{1}{n} \to 0 \) as \( n \to \infty \).
This example clearly shows the original statement is false, reinforcing the need for \( b_n \to 0 \) in any conditionally converging alternating series.