Alternating Series Test vs Ratio Test: when to use which?

📚 Quick Study Guide

• ➕ The Alternating Series Test (AST) is primarily used for series where terms alternate in sign. The general form is $\sum_{n=1}^{\infty} (-1)^n b_n$ or $\sum_{n=1}^{\infty} (-1)^{n+1} b_n$, where $b_n > 0$.
• ✅ To apply the AST, you need to verify two conditions:
  • 📉 $b_n$ is a decreasing sequence.
  • lim $b_n = 0$ as $n \to \infty$.
• ➗ The Ratio Test is best for series involving factorials or exponential terms. It examines the limit of the ratio of consecutive terms: $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$.
• 🚦 The Ratio Test has three possible outcomes:
  • If $L < 1$, the series converges absolutely.
  • If $L > 1$, the series diverges.
  • If $L = 1$, the test is inconclusive.
• 💡 When to choose: If you see alternating signs, think AST. If you see factorials or exponentials, think Ratio Test. If both apply or neither is obvious, consider other tests.
• 🚫 Important Note: The Ratio Test is inconclusive when the limit equals 1. In this case, other convergence tests are required.

Practice Quiz

1. Which test is most suitable for determining the convergence of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$?
   1. Alternating Series Test
   2. Ratio Test
   3. Integral Test
   4. Comparison Test
2. What condition MUST be met for the Alternating Series Test to be applicable?
   1. The terms must be positive.
   2. The terms must be decreasing and approach zero.
   3. The terms must be increasing.
   4. The terms must approach infinity.
3. For what type of series is the Ratio Test generally most effective?
   1. Series with alternating signs.
   2. Series involving factorials or exponentials.
   3. Series with decreasing terms.
   4. Series with increasing terms.
4. What conclusion can be drawn if applying the Ratio Test results in $L > 1$?
   1. The series converges absolutely.
   2. The series converges conditionally.
   3. The series diverges.
   4. The test is inconclusive.
5. What conclusion can be drawn if applying the Alternating Series Test shows that $\lim_{n \to \infty} b_n \neq 0$?
   1. The series converges absolutely.
   2. The series converges conditionally.
   3. The series diverges.
   4. The test is inconclusive.
6. Which test is most suitable for determining the convergence of the series $\sum_{n=1}^{\infty} \frac{n!}{n^n}$?
   1. Alternating Series Test
   2. Ratio Test
   3. Integral Test
   4. Comparison Test
7. What does it mean if the Ratio Test is inconclusive?
   1. The series converges.
   2. The series diverges.
   3. Another test must be used to determine convergence or divergence.
   4. The series oscillates.