How to Use the nth Term Test for Series: A Step-by-Step Approach

📚 Quick Study Guide

• ♾️ The nth Term Test is used to determine if an infinite series $\sum_{n=1}^{\infty} a_n$ *diverges*.
• 📝 The Test: If $\lim_{n \to \infty} a_n \neq 0$, 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 to use another test.
• 🧮 In simpler terms, if the terms of the series don't approach zero, the sum of those terms will grow without bound.
• 💡 The nth Term Test is usually the *first* test to apply because it's often the easiest.

🧪 Practice Quiz

1. Question 1: What is the conclusion of the nth Term Test if $\lim_{n \to \infty} a_n = 5$?
   1. Converges
   2. Diverges
   3. Inconclusive
   4. Oscillates
2. Question 2: Consider the series $\sum_{n=1}^{\infty} \frac{n}{2n+1}$. What is $\lim_{n \to \infty} a_n$?
   1. 0
   2. $\frac{1}{2}$
   3. 1
   4. $\infty$
3. Question 3: Does the series $\sum_{n=1}^{\infty} \frac{n}{2n+1}$ converge or diverge according to the nth Term Test?
   1. Converges
   2. Diverges
   3. Inconclusive
   4. Absolutely Converges
4. Question 4: What happens if $\lim_{n \to \infty} a_n = 0$ when using the nth Term Test?
   1. The series converges.
   2. The series diverges.
   3. The test is inconclusive.
   4. The series oscillates.
5. Question 5: For which of the following series does the nth Term Test *definitely* show divergence?
   1. $\sum_{n=1}^{\infty} \frac{1}{n^2}$
   2. $\sum_{n=1}^{\infty} \frac{1}{n}$
   3. $\sum_{n=1}^{\infty} \frac{n^2}{n}$
   4. $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$
6. Question 6: Find $\lim_{n \to \infty} a_n$ for the series $\sum_{n=1}^{\infty} \frac{n!}{2^n}$.
   1. 0
   2. 1
   3. 2
   4. $\infty$
7. Question 7: According to the nth Term Test, what can we conclude about the series $\sum_{n=1}^{\infty} \frac{n!}{2^n}$?
   1. Converges
   2. Diverges
   3. Inconclusive
   4. Absolutely Converges