📚 Topic Summary
The Divergence Test is a straightforward method used to determine if an infinite series diverges. The test states: if the limit of the sequence $a_n$ as $n$ approaches infinity is not equal to zero, then the series $\sum_{n=1}^{\infty} a_n$ diverges. However, if the limit is zero, the test is inconclusive, and another test must be used. The key is that if the terms of the series don't approach zero, the series cannot converge. It's one of the first tests you should consider when checking the convergence or divergence of a series. Remember, the test can only prove divergence; it cannot prove convergence.
In simpler terms: If the individual pieces you're adding up don't get super tiny, the whole thing will never settle down to a finite number. 🤯
🧮 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Divergence Test |
A. A series that does not converge. |
| 2. Limit |
B. The value that a function or sequence approaches as the input approaches some value. |
| 3. Infinite Series |
C. $\lim_{n \to \infty} a_n \neq 0$ |
| 4. Divergent Series |
D. The sum of infinitely many terms. |
| 5. Divergence Condition |
E. A test that states if the limit of the sequence does not equal zero, the series diverges. |
✏️ Part B: Fill in the Blanks
The Divergence Test is used to determine if a series _____. If the _____ of the sequence $a_n$ as $n$ approaches infinity is not equal to _____, then the series diverges. If the limit is zero, the test is _____. Therefore, other _____ tests should be performed.
🤔 Part C: Critical Thinking
Explain in your own words why the Divergence Test can only prove divergence, but not convergence. Provide an example to support your explanation.