Question

Describe how to apply the Alternating Series Test.

Short answer

Question: Explain how to apply the Alternating Series Test to determine the convergence or divergence of an alternating series. Answer: To apply the Alternating Series Test, follow these steps: 1. Identify an alternating series of the form \(\sum_{n=1}^{\infty} (-1)^n a_n\) or \(\sum_{n=1}^{\infty} (-1)^{n-1} a_n\), where the terms \(a_n\) are non-negative. 2. Ensure that the series' terms \(a_n\) are non-increasing and non-negative. 3. Apply the test by finding the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} a_n\). 4. Determine convergence or divergence: - If the limit is zero (i.e., \(\lim_{n \to \infty} a_n = 0\)), the series converges. - If the limit does not equal zero or does not exist, the series diverges.

Step-by-step solution

1. Identify the Alternating Series

An alternating series has a general form of: \(\sum_{n=1}^{\infty} (-1)^n a_n\) or \(\sum_{n=1}^{\infty} (-1)^{n-1} a_n\) where the terms \(a_n\) are non-negative. In mathematical notation, it can be described as \(a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + ... \)

2. Check the Series' Terms for Non-Increasing and Non-Negative

To apply the Alternating Series Test, the terms of the series \(a_n\) must be non-negative (i.e., \(a_n \ge 0\) for all \(n\)) and non-increasing (i.e., \(a_n \ge a_{n+1}\) for all \(n\)). Ensure that the two conditions are satisfied before applying the test to the series.

3. Apply the Alternating Series Test

Once the alternating series and its terms meet the conditions, apply the Alternating Series Test by taking the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} a_n\)

4. Determine Convergence or Divergence

Evaluate the limit obtained in step 3. - If the limit is zero, i.e., \(\lim_{n \to \infty} a_n = 0\), then the alternating series converges. - If the limit does not equal zero or does not exist, the alternating series diverges. After these steps, you can conclude whether the given alternating series converges or diverges based on the result of the Alternating Series Test.