When can you use the Alternating Series Test? Conditions for AST

📚 Quick Study Guide

🔢 The Alternating Series Test (AST) is used to determine the convergence of an alternating series. ➕ An alternating series has terms that alternate in sign, such as $\sum_{n=1}^{\infty} (-1)^n a_n$ or $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$, where $a_n > 0$ for all $n$. 📝 To apply the AST, two conditions must be met:
• 📉 Condition 1: The sequence $a_n$ must be decreasing, i.e., $a_{n+1} \le a_n$ for all $n$ greater than some integer $N$.
• 0️⃣ Condition 2: $\lim_{n \to \infty} a_n = 0$.
✅ If both conditions are satisfied, the alternating series converges. ❗ If either condition is not met, the AST is inconclusive, and you must use another test to determine convergence or divergence. 💡 The absolute value of the first term can give an upper bound for the error when approximating the infinite sum by a partial sum.

Practice Quiz

1. Question 1: Which of the following is a necessary condition for the Alternating Series Test to be applicable?
   1. The terms of the series must be positive.
   2. The terms of the series must be decreasing.
   3. The terms of the series must be increasing.
   4. The terms of the series must be constant.
2. Question 2: Consider the series $\sum_{n=1}^{\infty} (-1)^n \frac{1}{n}$. Does this series satisfy the conditions of the Alternating Series Test?
   1. No, it does not satisfy either condition.
   2. No, it only satisfies the decreasing condition.
   3. No, it only satisfies the limit condition.
   4. Yes, it satisfies both conditions.
3. Question 3: Suppose you have an alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} a_n$. What does the Alternating Series Test conclude if $\lim_{n \to \infty} a_n \neq 0$?
   1. The series converges conditionally.
   2. The series converges absolutely.
   3. The series diverges.
   4. The test is inconclusive.
4. Question 4: Which of the following series can have the Alternating Series Test directly applied to determine convergence?
   1. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   2. $\sum_{n=1}^{\infty} (-1)^n \frac{n}{n+1}$
   3. $\sum_{n=1}^{\infty} (-1)^n$
   4. $\sum_{n=1}^{\infty} \frac{1}{n}$
5. Question 5: For an alternating series to converge by the Alternating Series Test, the absolute value of the terms must approach what value as $n$ approaches infinity?
   1. 1
   2. $\infty$
   3. 0
   4. Any finite value
6. Question 6: Is the series $\sum_{n=1}^{\infty} (-1)^n \frac{n+1}{n}$ convergent or divergent based on the Alternating Series Test?
   1. Convergent
   2. Divergent
   3. The test is inconclusive
   4. Convergent absolutely
7. Question 7: Which condition is NOT part of the Alternating Series Test?
   1. $a_n > 0$ for all $n$.
   2. $\lim_{n \to \infty} a_n = 0$.
   3. $a_{n+1} \le a_n$ for all $n$.
   4. The terms must be positive.