Test Your Knowledge: Convergence and Divergence of Sequences

📚 Quick Study Guide

🔍 A sequence is a list of numbers in a specific order, often defined by a formula. 📈 Convergence means the sequence approaches a finite limit as the index approaches infinity. 📉 Divergence means the sequence does not approach a finite limit; it may oscillate or increase/decrease without bound. 🔢 Key limits to remember:
• $lim_{n \to \infty} \frac{1}{n^p} = 0$ for $p > 0$
• $lim_{n \to \infty} r^n = 0$ for $|r| < 1$
🧮 L'Hôpital's Rule can be used to evaluate limits of sequences when they take indeterminate forms. 🧪 Common convergence tests include the ratio test, root test, and comparison test. 💡 Monotonic sequences (either increasing or decreasing) that are bounded are guaranteed to converge.

Practice Quiz

1. What does it mean for a sequence to converge?
   1. It oscillates between two values.
   2. It approaches a finite limit.
   3. It increases without bound.
   4. It decreases without bound.
2. Which of the following sequences converges?
   1. $a_n = (-1)^n$
   2. $a_n = n^2$
   3. $a_n = \frac{1}{n}$
   4. $a_n = sin(n)$
3. The sequence $a_n = \frac{n}{n+1}$ converges to which value?
   1. 0
   2. 1
   3. -1
   4. $\infty$
4. Which test is most suitable for determining the convergence of the sequence $a_n = \frac{n!}{n^n}$?
   1. Comparison Test
   2. Ratio Test
   3. Root Test
   4. Integral Test
5. If a sequence is monotonically increasing and bounded above, it must:
   1. Diverge to infinity.
   2. Diverge to negative infinity.
   3. Converge.
   4. Oscillate.
6. What is the limit of the sequence $a_n = (1 + \frac{1}{n})^n$ as $n$ approaches infinity?
   1. 0
   2. 1
   3. $e$
   4. $\infty$
7. Which of the following statements is true about a divergent sequence?
   1. It always approaches a finite limit.
   2. It never oscillates.
   3. It does not approach a finite limit.
   4. It is always bounded.