Calculus Series Divergence Test Quiz: Assess Your Understanding

📚 Quick Study Guide

• 🔢 The Divergence Test states that if $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ diverges.
• ♾️ If $\lim_{n \to \infty} a_n = 0$, the Divergence Test is inconclusive; another test is needed.
• ⚠️ The Divergence Test can *only* prove divergence, not convergence.
• 💡 Remember to always check the limit as $n$ approaches infinity of the sequence $a_n$.
• 📝 Common limits to remember: $\lim_{n \to \infty} \frac{1}{n} = 0$, $\lim_{n \to \infty} c = c$ (where c is a constant).

🧪 Practice Quiz

1. Question 1: Determine whether the series $\sum_{n=1}^{\infty} \frac{n}{3n+1}$ diverges.
1. Converges
2. Diverges
3. Inconclusive
4. Absolutely Converges
2. Question 2: Determine whether the series $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$ diverges using the Divergence Test.
1. Diverges
2. Converges
3. Inconclusive
4. Absolutely Converges
3. Question 3: Determine whether the series $\sum_{n=1}^{\infty} \frac{n^2}{n^2+1}$ diverges.
1. Converges
2. Diverges
3. Inconclusive
4. Absolutely Converges
4. Question 4: What does the Divergence Test tell us about the series $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$?
1. The series diverges.
2. The series converges.
3. The test is inconclusive.
4. The series absolutely converges.
5. Question 5: Determine whether the series $\sum_{n=1}^{\infty} \frac{2n+3}{5n+4}$ diverges.
1. Converges
2. Diverges
3. Inconclusive
4. Absolutely Converges
6. Question 6: What is the first step in applying the Divergence Test to the series $\sum_{n=1}^{\infty} a_n$?
1. Compute the partial sums.
2. Find $\lim_{n \to \infty} a_n$.
3. Compare it to a known series.
4. Integrate the terms.
7. Question 7: For what type of series is the Divergence Test most useful?
1. Convergent series
2. Series where the terms approach zero
3. Series where the terms do not approach zero
4. Absolutely convergent series