Question

The following series do not satisfy the hypotheses of the alternating series test as stated. In each case, state which hypothesis is not satisfied. State whether the series converges absolutely.\(1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{1}{9}+\cdots\)

Short answer

The series doesn't have alternating signs; it diverges.

Step-by-step solution

Identify the Series

We are given the series: \(1 + \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots\). This series is not strictly alternating since terms don't consistently switch between positive and negative.

Check Alternating Series Test Hypotheses

For a series to satisfy the alternating series test, it must have terms that alternate in sign, and the absolute value of terms must be monotonically decreasing, tending to zero. Our series does not alternate in sign regularly because consecutive positive terms occur.

Analyze Monotonicity and Convergence

The term sequence \(|a_n| = \left\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{5}, \frac{1}{7}, \cdots \right\}\) is not strictly decreasing, violating one of the hypotheses of the alternating series test.

Check for Absolute Convergence

Calculate the absolute series: \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\). This is the harmonic series, which is known to diverge. Hence, the original series does not converge absolutely.

Key concepts

Monotonicity in Series

The concept of monotonicity in a series is crucial for applying tests such as the Alternating Series Test. In the context of series, monotonicity typically refers to a sequence where the terms are either non-increasing or non-decreasing. For a series to apply the Alternating Series Test, the absolute values of terms should form a monotonically decreasing sequence, converging to zero.

This means that each term in the sequence should be less than or equal to the previous term. In this way, the terms steadily approach zero, helping establish the convergence behavior expected by the criteria.

• **Alternating Series Test:** An alternating series, like \(-1^{n}a_n\), where \(|a_n|\) decreases monotonically and tends to zero, converges.
• **Check:** In our example series, the absolute term sequence \(|a_n| = \{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{5}, \frac{1}{7}, \cdots\}\) does not consistently decrease, disrupting monotonicity.

Thus, without satisfying monotonicity, our series can't efficiently use the Alternating Series Test to test convergence.

Absolute Convergence

Absolute convergence is a stronger form of convergence that applies to series where the sum of the absolute values of terms is convergent. If the series formed by taking the absolute values of each of the terms of the original series converges, then the original series also converges.

This is an important criterion because it implies the original series converges regardless of the order of its terms. However, if the series of absolute values diverges, then the original series isn't necessarily divergent, but it certainly doesn't converge absolutely.

• **Harmonic Series:** The absolute version of our series is the harmonic series, which is known to diverge.
• **Original Series:** Due to the divergence of the absolute series \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\), the given series does not converge absolutely.

Without absolute convergence, we have less certainty on the series' overall convergence without further tests.

Harmonic Series Convergence

The harmonic series is one of the classic examples in calculus and analysis, famous for its divergence despite the decreasing nature of its terms. It is represented as:

\[H = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\]
The harmonic series is a cornerstone in study due to its simplicity and its non-obvious divergence.

• **Divergence:** Each term decreases as expected, but their sum grows without bound.
• **Comparison Test:** The harmonic series diverges because it can be compared to a simpler divergent series of the form \(\frac{1}{2} + \frac{1}{2} + \cdots\), providing a basis to assess other series.

Understanding the divergence of harmonic series helps in recognizing the limits of convergence tests, focusing importance on proper term comparison and test criteria to determine series behavior effectively.