What is the Alternating Series Test? Definition and criteria

📚 Quick Study Guide

• ➕ An alternating series has terms that alternate in sign (positive, negative, positive, etc.).
• 🔢 It can be generally represented as $\sum_{n=1}^{\infty} (-1)^{n-1}b_n$ or $\sum_{n=1}^{\infty} (-1)^{n}b_n$, where $b_n > 0$ for all $n$.
• ✅ The Alternating Series Test states that if $b_n$ is a positive sequence that is decreasing (i.e., $b_{n+1} \le b_n$ for all $n$) and $\lim_{n \to \infty} b_n = 0$, then the alternating series converges.
• ❗ If $\lim_{n \to \infty} b_n \neq 0$, then the alternating series diverges (by the Divergence Test).
• 🧐 Be careful! The Alternating Series Test only tells you if the series converges; it doesn't tell you what it converges to.
• 💡 Remember to always check the limit of the absolute value of the terms first! It's a fast way to rule out divergence.

Practice Quiz

1. Which of the following series is an alternating series?

   1. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   2. $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$
   3. $\sum_{n=1}^{\infty} \frac{1}{2^n}$
   4. $\sum_{n=1}^{\infty} \frac{n}{n+1}$

2. What are the two conditions required for the Alternating Series Test to show convergence of $\sum_{n=1}^{\infty} (-1)^{n-1}b_n$?

   1. $b_n$ must be increasing and $\lim_{n \to \infty} b_n = 0$
   2. $b_n$ must be decreasing and $\lim_{n \to \infty} b_n = 0$
   3. $b_n$ must be constant and $\lim_{n \to \infty} b_n = 1$
   4. $b_n$ must be positive and $\lim_{n \to \infty} b_n = \infty$

3. Does the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ converge or diverge?

   1. Converges
   2. Diverges
   3. The test is inconclusive
   4. Cannot be determined

4. What is the first step you should take when assessing convergence or divergence of an alternating series?

   1. Apply the Ratio Test
   2. Check if the terms are decreasing
   3. Check if $\lim_{n \to \infty} b_n = 0$
   4. Apply the Integral Test

5. Which of the following series diverges by the Alternating Series Test?

   1. $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$
   2. $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^3}$
   3. $\sum_{n=1}^{\infty} (-1)^n \frac{n}{2n+1}$
   4. $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}$

6. If $\lim_{n \to \infty} b_n = 5$ for an alternating series $\sum_{n=1}^{\infty} (-1)^{n-1}b_n$, what can you conclude?

   1. The series converges
   2. The series converges conditionally
   3. The series diverges
   4. The series converges absolutely

7. Consider the series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n^p}$. For what values of $p$ does this series converge?

   1. $p > 1$
   2. $p \ge 1$
   3. $p < 1$
   4. $p \le 1$