Problem 21 - Thomas Calculus

Question

Which of the series Converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+3} $$

Step-by-Step Solution

Answer

The series converges conditionally but not absolutely.

Step 1: Identify the Series Type

The given series is \( \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+3} \). This is an alternating series because the term \((-1)^n\) causes the terms to switch sign at each step.

Step 2: Absolute Convergence Test

To check for absolute convergence, we test the series \( \sum_{n=1}^{\infty} \left| (-1)^n \frac{1}{n+3} \right| = \sum_{n=1}^{\infty} \frac{1}{n+3} \), which resembles a harmonic series. Since the harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is known to diverge, this series also diverges.

Step 3: Conditional Convergence Test

Next, we check for conditional convergence using the Alternating Series Test. The series \( \sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+3} \) converges if the terms \( \frac{1}{n+3} \) are positive, decreasing, and approach zero as \( n \to \infty \). Since \( \frac{1}{n+3} > 0 \) for all \( n \), \( \frac{1}{n+3} \) is decreasing, and \( \lim_{n \to \infty} \frac{1}{n+3} = 0 \), the series converges conditionally.

Key Concepts

Alternating Series Test

When working with series, particularly those that alternate between positive and negative terms, the Alternating Series Test is a practical tool. This test helps determine if an alternating series converges. It is applicable when a series has terms of the form \((-1)^n a_n\), where \(a_n\) is a positive sequence.

The test requires checking three conditions:

• The terms \(a_n\) must be positive.
• The sequence \(a_n\) should be eventually decreasing, meaning that each term is smaller than the preceding term for all sufficiently large \(n\).
• Most importantly, \(\lim_{n \to \infty} a_n = 0\).

If all three conditions are satisfied, the alternating series converges. In the given series \(\sum_{n=1}^{\infty}(-1)^{n} \frac{1}{n+3}\), these conditions hold true, and thus the series converges. This convergence is known as conditional, which we will explore further.

Absolute Convergence

Absolute convergence means a series converges when all its terms are made positive. To test for absolute convergence, you take the absolute value of each term in the series. For the series given, this becomes \(\sum_{n=1}^{\infty} \frac{1}{n+3}\).

This resembles the harmonic series \(\sum_{n=1}^{\infty} \frac{1}{n}\), which is known to diverge. The harmonic series diverges because its terms decrease very slowly, never quite approaching zero fast enough to sum up to a finite number.

Since the series \(\sum_{n=1}^{\infty} \frac{1}{n+3}\) also diverges in the same way, we conclude that the original given series does not converge absolutely. This lack of absolute convergence differentiates it from some other series that might see both positive and negative modifications still leading to convergence.

Conditional Convergence

Conditional convergence occurs when a series converges only because of the alternating nature of its terms. In the context of our series example, while the alternating series \(\sum_{n=1}^{\infty}(-1)^n \frac{1}{n+3}\) converges, its absolute counterpart \(\sum_{n=1}^{\infty} \frac{1}{n+3}\) does not.

Despite the absolute series diverging, the given series converges due to the alternating signs that help the terms cancel each other out more effectively. This balance achieved by alternating series is why the Alternating Series Test is vital in recognizing conditional convergence.

Understanding this difference equips you to distinguish between different types of convergent series and helps predict how alterations in series terms can change its overall behavior. Conditional convergence highlights the nuanced nature of infinite summations, offering insight that aids in deeper mathematical applications and understanding.