📚 Topic Summary
In calculus, a sequence is called monotonic if it is either entirely non-increasing or entirely non-decreasing. Essentially, it's like a one-way street – the terms either keep going up (increasing) or keep going down (decreasing). A sequence is bounded if all its terms are between two specific numbers. Think of it as the sequence being trapped within upper and lower limits. Understanding these properties is crucial for determining if a sequence converges (approaches a specific value) or diverges (goes off to infinity or oscillates).
🧮 Part A: Vocabulary
Match the terms with their definitions:
| Term |
Definition |
| 1. Monotonic Sequence |
A. A sequence whose terms are all greater than or equal to some number. |
| 2. Bounded Sequence |
B. A sequence that is either non-increasing or non-decreasing. |
| 3. Upper Bound |
C. A sequence whose terms are all less than or equal to some number. |
| 4. Lower Bound |
D. A number greater than or equal to all terms in the sequence. |
| 5. Convergent Sequence |
E. A sequence whose terms approach a specific value as the index approaches infinity. |
Matching Answers:
- 🔢 1 - B
- 📈 2 - D & A
- 📉 3 - C
- 📊 4 - A
- ✅ 5 - E
✍️ Part B: Fill in the Blanks
A sequence {$a_n$} is said to be ________ if $a_n \le a_{n+1}$ for all n, or if $a_n \ge a_{n+1}$ for all n. If there exists a number M such that $|a_n| \le M$ for all n, then the sequence is ________. A sequence that is both monotonic and bounded is guaranteed to ________.
Fill in the Blanks Answers:
- ♾️ monotonic
- 🔒 bounded
- 🎯 converge
🤔 Part C: Critical Thinking
Explain, in your own words, why a sequence that is increasing but not bounded above must diverge. Give a specific example of such a sequence.
Example Answer:
- 💡 If a sequence is increasing, it means each term is larger than the previous one. If it's not bounded above, there's no limit to how large the terms can get. Therefore, the terms will continue to grow without approaching any specific value, meaning the sequence diverges. An example is the sequence {$n$}, where each term is just the index number (1, 2, 3, ...). This sequence increases indefinitely and has no upper bound, so it diverges.