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.Let \(a_{n}^{+}=a_{n}\) if \(a_{n} \geq 0\) and \(a_{n}^{-}=-a_{n}\) if \(a_{n}<0 .\) (Also, \(a_{n}^{+}=0\) if \(a_{n}<0\) and \(a_{n}^{-}=0\) if \(\left.a_{n} \geq 0 .\right)\) If \(\sum_{n=1}^{\infty} a_{n}\) converges conditionally but not absolutely, then neither \(\sum_{n=1}^{\infty} a_{n}^{+}\) nor \(\sum_{n=1}^{\infty} a_{n}^{-}\) converge.

Short answer

True; neither \( \sum_{n=1}^{\infty} a_n^+ \) nor \( \sum_{n=1}^{\infty} a_n^- \) converge if \( \sum_{n=1}^{\infty} a_n \) converges conditionally.

Step-by-step solution

Understand the Definitions

The sequence \( a_n^+ \) includes the positive parts of \( a_n \) and is zero when \( a_n < 0 \). The sequence \( a_n^- \) includes the negative parts (made positive) of \( a_n \) and is zero when \( a_n \geq 0 \). In conditional convergence, the series \( \sum_{n=1}^{\infty} a_n \) converges but \( \sum_{n=1}^{\infty} |a_n| \) diverges.

Analyze Conditional Convergence

If \( \sum_{n=1}^{\infty} a_n \) converges conditionally, then the positive and negative parts of the series balance each other to converge, but the absolute series \( \sum_{n=1}^{\infty} |a_n| \) diverges because individually either \( \sum_{n=1}^{\infty} a_n^+ \) or \( \sum_{n=1}^{\infty} a_n^- \) should diverge.

Consider \( \sum_{n=1}^{\infty} a_n^+ \) and \( \sum_{n=1}^{\infty} a_n^- \) Separately

Since \( a_n \)’s conditionally converge, the positive and negative series' partial sums do not converge individually, because if either part converged, the other would only contribute a finite amount, making \( \sum_{n=1}^{\infty} a_n \) converge absolutely, which contradicts conditional convergence.

Provide a Counterexample If Possible

We cannot provide a counterexample where both component series \( \sum_{n=1}^{\infty} a_n^+ \) or \( \sum_{n=1}^{\infty} a_n^- \) converge in a conditionally convergent series, as that would imply that the original series \( \sum_{n=1}^{\infty} a_n \) should converge absolutely.

Key concepts

Series Convergence

Series convergence is an essential concept in mathematical analysis. When we examine an infinite series \( \sum_{n=1}^{\infty} a_n \), we are interested in whether the sum approaches a certain finite value as more and more terms are added. This is what we call convergence. If the series does not approach a finite limit, it is said to diverge.
To determine if a series converges, we often use tests such as the ratio test, root test, or alternating series test. These tests help us analyze the behavior of the terms in the series as \( n \) becomes very large. If a series converges, we can think of it as reaching stability, meaning the sum gets closer to a fixed number.
Conditional convergence occurs when a series \( \sum_{n=1}^{\infty} a_n \) converges, but the series of its absolute values \( \sum_{n=1}^{\infty} |a_n| \) diverges. This implies a special balance between positive and negative terms in the series, leading to convergence without absolute convergence.
Understanding series convergence allows us to better comprehend how infinite processes can result in finite outcomes. It is essential in areas like calculus, and helps with evaluating series solutions to complex mathematical problems.

Absolute Convergence

Absolute convergence is a stronger form of series convergence. A series \( \sum_{n=1}^{\infty} a_n \) is absolutely convergent if the series of its absolute values \( \sum_{n=1}^{\infty} |a_n| \) also converges. Absolute convergence not only implies that the series itself converges but also that it would converge under any reordering of its terms—a powerful property that conditional convergence does not have.
For absolute convergence, every term in the series contributes positively to the overall sum when considering their absolute values, ensuring that the overall sum stabilizes without any cancellation between positive and negative parts.
To test for absolute convergence, mathematicians often use comparison tests, such as comparing the series to a known convergent series. If \( \sum_{n=1}^{\infty} a_n \) is absolutely convergent, it enjoys a more stable and predictable convergence behavior. It overrides issues related to alternating positive and negative terms, which can sometimes cause conditionally convergent series to delicately balance in a way that makes them less robust.

Positive and Negative Parts of Series

When dealing with series, particularly those that converge conditionally, separating terms into positive and negative components can be insightful. This involves defining \( a_n^+ \) and \( a_n^- \), representing the positive and negative parts of the sequence \( a_n \), respectively.

• \( a_n^+ = a_n \) if \( a_n \geq 0 \), and zero otherwise.
• \( a_n^- = -a_n \) if \( a_n < 0 \), and zero otherwise.

The positive series \( \sum_{n=1}^{\infty} a_n^+ \) comprises all positive contributions, while the negative series \( \sum_{n=1}^{\infty} a_n^- \) consists of all negative contributions flipped to positive.
In a conditionally convergent series, neither of these partial series converges on their own. If either \( \sum_{n=1}^{\infty} a_n^+ \) or \( \sum_{n=1}^{\infty} a_n^- \) were to converge separately, it would imply that the entire series could converge absolutely, contradicting the nature of conditional convergence where the overall opposition between positive and negative terms ensures the convergence of the entire series.