Question

\(1+\frac{1}{2^{2}}-\frac{1}{3^{2}}-\frac{1}{4^{2}}+\frac{1}{5^{2}}+\frac{1}{6^{2}}-\frac{1}{7^{2}}-\frac{1}{8^{2}}+\ldots \cdots\) (a) conditionally convergent (b) convergent (c) oscillatory (d) divergent.

Short answer

The series is conditionally convergent.

Step-by-step solution

Understand the Sequence

The given problem is the series \(1+\frac{1}{2^{2}}-\frac{1}{3^{2}}-\frac{1}{4^{2}}+\frac{1}{5^{2}}+\frac{1}{6^{2}}-\frac{1}{7^{2}}-\frac{1}{8^{2}}+\ldots\). It has alternating positive and negative terms. Each pair of terms such as \(+\frac{1}{2^{2}}-\frac{1}{3^{2}}\) suggests a pattern that alternates signs every two terms.

Consider the Nature of Terms

The series is composed of positive and negative terms in a specific pattern predominated by the oscillation groups of two terms. This suggests it is neither purely positive nor purely negative, hence it is not strictly oscillatory.

Analyze Convergence Criteria

To determine convergence, analyze the behavior of the terms. Each pair acts like a telescopic series, where the net effect of the pair is smaller with progressing indices. The individual terms decrease monotonically and approach zero.

Apply the Alternating Series Test

The alternating series test requires that the absolute value of terms decrease monotonically and approach zero. In this series, the absolute values \(\frac{1}{n^2}\) decrease and tend to zero as \(n\) approaches infinity, satisfying the test conditions for convergence.

Determine Type of Convergence

Given that the absolute values \(\frac{1}{n^2}\) create a convergent series when summed alone (the series \(\sum \frac{1}{n^2}\) is known to converge), and the alternating nature fits the alternating series test, the series converges absolutely.

Key concepts

Convergence

Convergence refers to the behavior of a series or sequence as it approaches a specific value or condition as more terms are added. When we talk about the convergence of the series, we are interested in whether the sum of its infinite number of terms approaches a particular number.
In the exercise given, we analyze the series, where the absolute values of its terms decrease and their overall sum stabilizes. In simpler terms, no matter how many terms we add from this sequence, they tend to get us closer to a fixed number, rather than spiraling off to infinity or having no particular convergence pattern.
This is an important characteristic because it ensures that adding more terms doesn’t lead to wild variations or infinite results.

Alternating Series Test

The Alternating Series Test (or Leibniz Test) is a vital tool to check the convergence of an alternating series. An alternating series is characterized by terms whose signs alternate between positive and negative. To apply this test, two main conditions must be satisfied:

• The absolute value of the terms must decrease monotonically. This means that each term's absolute value is smaller than or equal to the term before it. For instance, \(\frac{1}{3^2} > \frac{1}{4^2}\).
• The absolute value of the terms must approach zero as the series progresses. This implies that as you progress through more terms, each term contributes less and less to the total sum.

In the exercise, this test confirms that the series converges because the sequence's terms follow the conditions set by the test.

Absolute Convergence

Absolute convergence is a stronger form of convergence in series analysis. A series is said to converge absolutely if the series formed by taking the absolute value of each term also converges. This means that even when we ignore the alternating aspects and just consider the positive magnitudes of the terms, the series still converges.
For the series in the exercise, the absolute series \(\sum \frac{1}{n^2}\) converges. This is a well-known result in harmonic series studies, where each term contributes positively to a sum that does not extend to infinity. Because of this convergence in the absolute sense, we can safely say the series converges absolutely.
Absolute convergence implies regular convergence and ensures stability in the behavior of the sum of the series.

Series Analysis

Series analysis involves determining the behavior and properties of a series through various tests and criteria. By breaking down the series into its parts, we assess its convergence, type, and other characteristics using analytical tools like the Alternating Series Test and absolute convergence criteria.
This analytical approach provides insight into both the series' individual terms and its overall behavior. It identifies whether the series will converge or diverge, and if it's condensing towards a specific limit or value.
Such analysis is crucial for understanding complex series, especially those with alternating signs, as it helps determine practical applications and predictions in various fields like calculus and complex analysis.