ยง11.4: The Comparison Tests
This section focuses on comparing a positive-term series to a known benchmark series, usually a geometric series or a $p$-series. You should learn how to use both the Direct Comparison Test and the Limit Comparison Test, choose a sensible comparison series, and write a correct conclusion.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| OpenStax Calculus: Comparison Tests | Textbook Notes | A strong textbook-style overview of the comparison test and limit comparison test. This is a good place to review the statements of the tests and see how they connect to familiar benchmark series such as geometric series and $p$-series. | Read Notes |
| Paul's Notes: Comparison / Limit Comparison Test | Study Guide | Especially helpful for learning the decision-making process. Use this when you want practice deciding whether a direct comparison is enough or whether the limit comparison test is the cleaner choice. | Read Notes |
| Paul's Practice: Comparison Test / Limit Comparison Test | Practice Set | A solid practice bank for building fluency with the algebra in these problems. Focus on identifying the dominant term and matching the series to an appropriate benchmark. | Practice |
| Paul's Notes: Strategy for Series | Decision Guide | Useful once you have seen several convergence tests. For this section, the most important idea is that series resembling geometric series or $p$-series are often good candidates for direct or limit comparison. | Open Guide |
| OpenStax Key Concepts (Chapter 5) | Quick Reference | A concise chapter-level summary for a fast reminder of the main facts before homework, quiz practice, or cumulative review. | Open Summary |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| Comparison Tests | University of Houston | A University of Houston video resource on the comparison tests. A good option if you want another explanation of the main ideas and a different presentation style from the textbook notes. | Watch |
| Full Lecture: Comparison Test + Limit Comparison Test | Professor Leonard | A strong full-length lecture if you want a slower, more complete explanation of both tests. It carefully develops the logic behind the comparison tests and works through examples in a classroom style. | Watch |
| Direct Comparison Test | The Organic Chemistry Tutor | A focused refresher on the Direct Comparison Test. Watch this if you want extra help choosing an inequality in the correct direction and comparing to a familiar benchmark series. | Watch |
| Limit Comparison Test | The Organic Chemistry Tutor | A good short video on the mechanics of the Limit Comparison Test. It emphasizes how to choose a sensible comparison series, compute the ratio limit, and interpret what the resulting constant tells you. | Watch |
| Choosing Which Convergence Test to Apply | Dr. Trefor Bazett | This video is a little broader than just comparison tests, but it is very useful when you are unsure which convergence test to try. It can help you decide whether comparison is the right tool for the series in front of you. | Watch |