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.Suppose that \(a_{n}\) is a sequence such that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) converges for every possible sequence \(b_{n}\) of zeros and ones. Does \(\sum_{n=1}^{\infty} a_{n}\) converge absolutely?

Short answer

False; for example, \(a_n = (-1)^n / n\) is conditionally convergent.

Step-by-step solution

Understand the statement

We need to determine whether the series \(\sum_{n=1}^{\infty} a_{n}\) converges absolutely given that \(\sum_{n=1}^{\infty} a_{n} b_{n}\) converges for every sequence \(b_n\) consisting of zeros and ones. A series \(\sum_{n=1}^{\infty} a_n\) converges absolutely if \(\sum_{n=1}^{\infty} |a_n|\) also converges.

Analyze the given condition

Since \(\sum_{n=1}^{\infty} a_{n} b_{n}\) converges for every binary sequence \(b_n\), in particular, it must converge for the sequence where \(b_n = 1\) for all \(n\). In this case, \(\sum_{n=1}^{\infty} a_{n} b_{n}\) simply becomes \(\sum_{n=1}^{\infty} a_{n}\). Hence, \(\sum_{n=1}^{\infty} a_{n}\) must converge.

Explore absolute convergence

Absolute convergence requires \(\sum_{n=1}^{\infty} |a_{n}|\) to converge. Suppose \(a_n\) has terms that alternate in sign, e.g. \(a_1 = 1, a_2 = -1, a_3 = 1, a_4 = -1, \ldots\). In this case, \(\sum a_n\) could conceivably converge, but \(\sum |a_n| = 1 + 1 + 1 + \ldots\) would diverge. Therefore, we inquire if \(\sum a_n b_n\) could still converge depending on \(b_n\).

Construct a counterexample if necessary

Consider \(a_n = (-1)^n / n\), which is a conditionally convergent series (i.e., \(\sum_{n=1}^{\infty} a_n\) converges, but \(\sum_{n=1}^{\infty} |a_n|\) does not). For any sequence of zeros and ones, \(b_n\) would need to omit an infinite number of terms or specifically pick terms such that \(\sum a_n b_n\) converges.

Conclusion

The statement is false. The series \(\sum_{n=1}^{\infty} a_n\) does not necessarily converge absolutely, as demonstrated by the provided counterexample \(a_n = (-1)^n / n\).

Key concepts

Sequence Analysis

Sequence analysis involves the examination of the behavior and properties of sequences as they progress towards infinity. Consider a sequence \( a_n \), which consists of numbers arranged in a specific order. Analyzing this sequence, especially through convergence tests, allows us to understand whether the sequence approaches a specific value or pattern over time.

When engaging in sequence analysis, several critical things should be considered:

• Convergence: Whether the terms of the sequence approach a certain fixed value as the number of terms increases.
• Divergence: If the sequence does not approach any specific limit.
• Boundedness: Whether the sequence remains confined within certain bounds.

Understanding these elements is vital to decide if a series, which is an infinite sum of sequence terms, converges or diverges.

Absolute Convergence

Absolute convergence of a series \( \sum_{n=1}^{\infty} a_n \) implies that the series will converge even when all its terms are replaced by their absolute values. This is stronger than regular convergence, as it ensures that the rearrangement of terms does not affect the convergence of the series.

Mathematically, a series \( \sum_{n=1}^{\infty} a_n \) is said to converge absolutely if \( \sum_{n=1}^{\infty} |a_n| \) converges.

• Absolute convergence guarantees that the total sum approaches a consistent limit, regardless of the sequence's individual term order.
• Check for absolute convergence using tests like the Comparison Test or the Ratio Test.
• If a series converges absolutely, it also converges in the standard sense. However, not all convergent series converge absolutely.

The distinction is crucial because it affects how we can manipulate series in mathematical operations.

Conditionally Convergent Series

A conditionally convergent series is slightly trickier. This type of series converges in the standard sense, but does not converge absolutely. In other words, \( \sum_{n=1}^{\infty} a_n \) converges, but \( \sum_{n=1}^{\infty} |a_n| \) does not.

Consider the alternating series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \). This series is quite famous for being conditionally convergent:

• It converges due to the alternating behavior, having terms that decrease in magnitude and change signs.
• The sum of the absolute values, \( \sum_{n=1}^{\infty} \frac{1}{n} \), diverges (it's a harmonic series).

Understanding conditional convergence helps in predicting which operations can safely be performed on a series. These are pivotal in real-world applications and theoretical analysis, as they show the limitations of series rearrangement and manipulation.