Applying Alternating Series and Absolute Convergence Tests

Introduction

Key Concepts

1. Understanding Series and Convergence

In calculus, a series is the sum of the terms of a sequence. Formally, for a sequence \( \{a_n\} \), the series is denoted as: $$ \sum_{n=1}^{\infty} a_n $$ A series can either converge or diverge. A convergent series approaches a finite limit as the number of terms grows indefinitely, whereas a divergent series does not approach a finite limit.

2. Alternating Series

An alternating series is a series in which the signs of the terms alternate between positive and negative. It can be expressed in the general form: $$ \sum_{n=1}^{\infty} (-1)^{n+1} b_n \quad \text{or} \quad \sum_{n=1}^{\infty} (-1)^n b_n $$ where \( b_n > 0 \) for all \( n \). Alternating series often arise in practical applications where quantities alternate between gains and losses, such as electrical engineering and economics.

3. The Alternating Series Test (Leibniz's Test)

The Alternating Series Test, also known as Leibniz's Test, provides criteria to determine the convergence of an alternating series. A series \( \sum_{n=1}^{\infty} (-1)^{n+1} b_n \) converges if the following two conditions are met:

1. Monotonic Decrease: \( b_{n+1} \leq b_n \) for all \( n \).
2. Limit to Zero: \( \lim_{n \to \infty} b_n = 0 \).

If both conditions are satisfied, the alternating series converges. However, it's important to note that this test does not determine whether the convergence is absolute or conditional.

4. Absolute Convergence

A series \( \sum_{n=1}^{\infty} a_n \) is said to absolutely converge if the series of absolute values \( \sum_{n=1}^{\infty} |a_n| \) converges. Absolute convergence implies convergence of the original series, but the converse is not necessarily true. If a series converges absolutely, it is called absolutely convergent; otherwise, if it converges but does not converge absolutely, it is conditionally convergent.

5. The Absolute Convergence Test

The Absolute Convergence Test states that if a series \( \sum_{n=1}^{\infty} a_n \) converges absolutely, then it converges. To apply this test, one examines the convergence of the series \( \sum_{n=1}^{\infty} |a_n| \). If this series converges, the original series converges absolutely.

6. Comparison Between Alternating Series Test and Absolute Convergence Test

While both tests assess the convergence of a series, they serve different purposes. The Alternating Series Test is specifically designed for alternating series and can establish conditional convergence. In contrast, the Absolute Convergence Test assesses absolute convergence, which ensures convergence regardless of the series being alternating or not.

7. Practical Examples

Let's apply both tests to concrete examples to solidify understanding.

Example 1: Alternating Series Test

Consider the series: $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} $$ Here, \( b_n = \frac{1}{n} \).

1. **Monotonic Decrease:** \( b_{n+1} = \frac{1}{n+1} < \frac{1}{n} = b_n \) for all \( n \).

2. **Limit to Zero:** \( \lim_{n \to \infty} \frac{1}{n} = 0 \).

Since both conditions are satisfied, the series converges by the Alternating Series Test.

Example 2: Absolute Convergence Test

Consider the series: $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$

To test for absolute convergence, examine the series of absolute values: $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} = \sum_{n=1}^{\infty} n^{-1/2} $$

This is a p-series with \( p = \frac{1}{2} \). Since \( p \leq 1 \), the series diverges. Therefore, the original series does not converge absolutely. However, applying the Alternating Series Test:

1. Monotonic Decrease: \( b_{n+1} = \frac{1}{\sqrt{n+1}} < \frac{1}{\sqrt{n}} = b_n \) for all \( n \).
2. Limit to Zero: \( \lim_{n \to \infty} \frac{1}{\sqrt{n}} = 0 \).

Both conditions are met, so the series converges conditionally.

8. Convergence Criteria and Tests

Understanding different convergence tests is crucial for analyzing series. Beyond the Alternating Series and Absolute Convergence Tests, there are other methods such as the Ratio Test, Root Test, and Integral Test that assist in determining the behavior of series. Each test has its applicability depending on the form of the series in question.

9. Importance in Calculus BC

Mastering convergence tests is essential for solving complex calculus problems encountered in the AP Calculus BC curriculum. These tests not only help in determining the convergence of series but also in understanding the nature of functions represented by series expansions, which is fundamental in higher-level mathematics and its applications.

Comparison Table

Summary and Key Takeaways