How to Identify When to Use the nth Term Test Effectively

📚 Quick Study Guide

• 🔢 The nth Term Test is primarily used to determine if an infinite series $\sum_{n=1}^{\infty} a_n$ extbf{diverges}. It cannot prove convergence.
• 🎯 Test Condition: If $\lim_{n \to \infty} a_n \neq 0$ or if the limit does not exist, then the series $\sum_{n=1}^{\infty} a_n$ diverges.
• ⚠️ Important Note: If $\lim_{n \to \infty} a_n = 0$, the test is inconclusive. This means the series might converge or diverge, and you'll need another test to determine its behavior (e.g., Integral Test, Comparison Test, Ratio Test).
• 📝 Example: Consider $\sum_{n=1}^{\infty} \frac{n}{2n+1}$. Here, $\lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2} \neq 0$. Thus, the series diverges by the nth Term Test.
• 💡 Strategy: Always check the nth Term Test *first*. It's often the quickest way to rule out convergence, saving you time.

Practice Quiz

1. Which of the following is the primary purpose of the nth Term Test?

   1. (A) To determine if a series converges.
   2. (B) To determine if a series diverges.
   3. (C) To find the sum of a convergent series.
   4. (D) To find the rate of convergence of a series.

2. The nth Term Test states that if $\lim_{n \to \infty} a_n = L$ and $L \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$:

   1. (A) Converges.
   2. (B) Diverges.
   3. (C) Is inconclusive.
   4. (D) Converges to L.

3. What happens if $\lim_{n \to \infty} a_n = 0$ when applying the nth Term Test to the series $\sum_{n=1}^{\infty} a_n$?

   1. (A) The series converges.
   2. (B) The series diverges.
   3. (C) The test is inconclusive, and another test is needed.
   4. (D) The series converges to 0.

4. Consider the series $\sum_{n=1}^{\infty} \frac{n^2}{3n^2 + 1}$. What does the nth Term Test tell us about its convergence or divergence?

   1. (A) The series converges.
   2. (B) The series diverges.
   3. (C) The test is inconclusive.
   4. (D) The series converges to $\frac{1}{3}$.

5. For which of the following series can the nth Term Test be used to immediately conclude divergence?

   1. (A) $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   2. (B) $\sum_{n=1}^{\infty} \frac{1}{n}$
   3. (C) $\sum_{n=1}^{\infty} \frac{n}{n+1}$
   4. (D) $\sum_{n=1}^{\infty} \frac{1}{2^n}$

6. If $\lim_{n \to \infty} a_n$ does not exist, what can you conclude about the series $\sum_{n=1}^{\infty} a_n$?

   1. (A) The series converges.
   2. (B) The series diverges.
   3. (C) The test is inconclusive.
   4. (D) The series oscillates.

7. Why is the nth Term Test often the first test to apply when analyzing a series?

   1. (A) It always provides a definitive answer.
   2. (B) It is computationally simple and can quickly identify divergence.
   3. (C) It is required before applying any other convergence test.
   4. (D) It directly calculates the sum of the series.