Which Of The Following Series Are Conditionally Convergent

Which of the Following Series are Conditionally Convergent?

Conditional convergence, a fascinating concept in the realm of infinite series, describes a series that converges to a finite sum but diverges when the absolute values of its terms are considered. Understanding conditional convergence requires a firm grasp of convergence tests and a nuanced understanding of the behavior of infinite sums. This article will delve into the intricacies of conditional convergence, exploring various examples and providing a structured approach to determining whether a given series exhibits this unique property.

Understanding Convergence and Divergence

Before diving into conditional convergence, let's refresh our understanding of the basic types of convergence and divergence for infinite series:

1. Absolute Convergence: A series Σa_n is absolutely convergent if the series of the absolute values of its terms, Σ|a_n|, converges. If a series is absolutely convergent, it is also convergent. This is a crucial point; absolute convergence implies convergence.

2. Conditional Convergence: A series Σa_n is conditionally convergent if it converges, but the series of its absolute values, Σ|a_n|, diverges. This means the series converges only due to the cancellation of positive and negative terms; if we consider only the magnitudes of the terms, the series would blow up to infinity.

3. Divergence: A series Σa_n is divergent if it does not converge to a finite sum. The series's partial sums either increase or decrease without bound, oscillate without settling on a limit, or exhibit other erratic behavior.

Key Tests for Convergence and Divergence

Several tests help determine whether a series converges absolutely, converges conditionally, or diverges. Some of the most important include:

1. The Ratio Test: This test is particularly useful for series involving factorials or exponential terms. For a series Σa_n:

• If lim_n→∞ |a_n+1/a_n| < 1, the series converges absolutely.
• If lim_n→∞ |a_n+1/a_n| > 1, the series diverges.
• If lim_n→∞ |a_n+1/a_n| = 1, the test is inconclusive.

2. The Root Test: Similar to the ratio test, the root test examines the nth root of the absolute value of the terms:

• If lim_n→∞ √|a_n| < 1, the series converges absolutely.
• If lim_n→∞ √|a_n| > 1, the series diverges.
• If lim_n→∞ √|a_n| = 1, the test is inconclusive.

3. The Integral Test: This test compares the series to an integral. If f(x) is a positive, decreasing, and continuous function such that f(n) = a_n for all n, then:

• If ∫₁^∞ f(x) dx converges, the series Σa_n converges.
• If ∫₁^∞ f(x) dx diverges, the series Σa_n diverges.

4. The Comparison Test: This test compares the series to a known convergent or divergent series. If 0 ≤ a_n ≤ b_n for all n, and Σb_n converges, then Σa_n converges. Conversely, if 0 ≤ b_n ≤ a_n for all n, and Σb_n diverges, then Σa_n diverges.

5. The Alternating Series Test: This test applies specifically to alternating series (series with terms that alternate in sign). If a_n is a decreasing sequence of positive terms such that lim_n→∞ a_n = 0, then the alternating series Σ(-1)ⁿa_n converges. Note that this test only establishes convergence; it doesn't guarantee absolute convergence.

Identifying Conditionally Convergent Series: A Step-by-Step Approach

Let's outline a systematic approach to determine if a given series is conditionally convergent:

Step 1: Check for Absolute Convergence: First, apply appropriate tests (Ratio, Root, Integral, Comparison) to the series Σ|a_n|. If this series converges, then the original series Σa_n converges absolutely, and therefore it is not conditionally convergent.

Step 2: Check for Convergence: If the series Σ|a_n| diverges, we need to determine if the original series Σa_n converges. The alternating series test is often helpful here, particularly if the series alternates in sign. Other convergence tests might also be applicable.

Step 3: Conclude: If Σa_n converges (as confirmed in Step 2) and Σ|a_n| diverges (as confirmed in Step 1), then the series Σa_n is conditionally convergent.

Examples of Conditionally Convergent Series

Let's illustrate with some classic examples:

1. The Alternating Harmonic Series: Σ(-1)ⁿ⁺¹(1/n) = 1 - 1/2 + 1/3 - 1/4 + ...

This series converges by the alternating series test. However, the series of absolute values, Σ|(-1)ⁿ⁺¹(1/n)| = Σ(1/n), is the harmonic series, which is known to diverge. Therefore, the alternating harmonic series is conditionally convergent.

2. Σ(-1)ⁿ/√n: This series converges by the alternating series test. However, the series of absolute values, Σ1/√n, is a p-series with p = 1/2 < 1, which diverges. Thus, this series is conditionally convergent.

3. Σ(-1)ⁿ/(n ln n): For n > 1, the terms decrease in magnitude and approach zero. By the alternating series test, the series converges. The series of absolute values, Σ1/(n ln n), diverges by the integral test. Therefore, this series is conditionally convergent.

The Riemann Rearrangement Theorem and its Implications

A fascinating aspect of conditionally convergent series is the Riemann Rearrangement Theorem. This theorem states that the terms of a conditionally convergent series can be rearranged to converge to any real number, or even to diverge! This highlights the delicate balance between convergence and divergence in conditionally convergent series. The order of terms significantly impacts the sum. Absolutely convergent series do not exhibit this behavior; their sum remains invariant under rearrangement.

Conclusion

Conditional convergence represents a nuanced area within the study of infinite series. Understanding the difference between absolute and conditional convergence is critical for accurately analyzing the behavior of infinite sums. Mastering the various convergence tests and applying a structured approach, as outlined above, is crucial for determining whether a given series exhibits conditional convergence. The Riemann Rearrangement Theorem underscores the delicate nature of conditionally convergent series and their sensitivity to the order of their terms. By understanding these concepts, you can confidently navigate the fascinating world of infinite series and their often-surprising behavior.