The Divergence Test: Can It Prove Convergence? (Important Misconception)

📚 Quick Study Guide

• 🔍 The Divergence Test states: If $\lim_{n \to \infty} a_n \neq 0$, then the series $\sum_{n=1}^{\infty} a_n$ diverges.
• 💡 The converse is NOT true: If $\lim_{n \to \infty} a_n = 0$, we CANNOT conclude that the series $\sum_{n=1}^{\infty} a_n$ converges. The series may converge or diverge.
• 📝 Examples of series where $\lim_{n \to \infty} a_n = 0$ but the series diverge include the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$.
• ➕ Other convergence tests (Integral Test, Comparison Test, Ratio Test, Root Test, Alternating Series Test) are needed to determine convergence when $\lim_{n \to \infty} a_n = 0$.
• ➗ The Divergence Test can ONLY prove divergence; it can never prove convergence.

Practice Quiz

1. Question 1: The Divergence Test can be used to prove:
   1. A. Convergence of a series
   2. B. Divergence of a series
   3. C. Both convergence and divergence
   4. D. Neither convergence nor divergence
2. Question 2: According to the Divergence Test, if $\lim_{n \to \infty} a_n = 0$, then the series $\sum a_n$:
   1. A. Converges
   2. B. Diverges
   3. C. May converge or diverge
   4. D. Oscillates
3. Question 3: Which of the following series can be proven to diverge using the Divergence Test?
   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. $\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$
4. Question 4: If $\lim_{n \to \infty} a_n = 2$, what can you conclude about the series $\sum a_n$?
   1. A. The series converges to 2
   2. B. The series diverges
   3. C. The series converges to 0
   4. D. The series converges, but not necessarily to 0 or 2
5. Question 5: The harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ has the property that $\lim_{n \to \infty} \frac{1}{n} = 0$. What does this imply according to the Divergence Test?
   1. A. The series converges
   2. B. The series diverges
   3. C. The Divergence Test is inconclusive
   4. D. The series oscillates
6. Question 6: For which series is the Divergence Test most directly applicable and conclusive?
   1. A. $\sum_{n=1}^{\infty} \frac{1}{n^3}$
   2. B. $\sum_{n=1}^{\infty} e^{-n}$
   3. C. $\sum_{n=1}^{\infty} \frac{n}{2n+5}$
   4. D. $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^2}$
7. Question 7: What other test should you consider when the Divergence Test is inconclusive (i.e., limit is 0)?
   1. A. Only the Ratio Test
   2. B. Only the Root Test
   3. C. Any other convergence test (e.g., Integral, Comparison, Alternating Series)
   4. D. No other test is needed; the series converges