ยง11.3: Integral Test
This section focuses on using a related improper integral to determine whether a positive-term series converges or diverges. The main ideas are checking the hypotheses of the Integral Test, evaluating the matching improper integral, recognizing the $p$-series pattern, and understanding when the test is an efficient choice.
Library, Handouts & Tools
| Resource Name | Type | Description | Link |
|---|---|---|---|
| Comparison Tests Practice | Class Handout | For each series, choose an appropriate comparison series, decide whether to use the Comparison Test or the Limit Comparison Test, and determine whether the series converges or diverges. | Download PDF |
| OpenStax Calculus: Integral Test | Textbook Notes | Best single textbook-style overview here for the Integral Test: the hypotheses on $f(x)$, how the improper integral controls convergence, and how the test connects directly to classic $p$-series examples. | Read Notes |
| Paul's Notes: Integral Test | Study Guide | Especially useful for learning the workflow: define the matching function $f(x)$, verify that it is positive, continuous, and decreasing, then evaluate the improper integral and transfer that conclusion back to the series. | Read Notes |
| Paul's Practice: Integral Test | Practice Set | A focused set of integral-test problems with exactly the kind of examples students need: rational powers, radicals, and logarithmic denominators where the test is natural but the hypotheses still matter. | Practice |
| OpenStax Chapter 5 Exercises | Textbook Practice | Good mixed practice from the same section, including deciding whether the test applies, working with $p$-series, and estimating convergence or divergence through the related improper integral. | Open Section |
Video Lectures
| Topic | Source | Description | Link |
|---|---|---|---|
| The Integral Test | University of Houston | Lecture explaining when the Integral Test applies and how it connects to $p$-series. | Watch Video |
| Lecture 37: Infinite Series | MIT OCW | Use this when you want a fuller lecture treatment instead of a short clip. It places the Integral Test inside the bigger picture of infinite series and convergence tests, so you can see when this test is the right tool and how it fits with the rest of the chapter. | Watch Lecture |
| Using the Integral Test for Convergence/Divergence of Series | Professor Leonard | Choose this if you learn best from a longer classroom-style explanation. You will see the hypotheses checked carefully, the improper integral evaluated step by step, and the conclusion tied back to whether the series converges or diverges. | Watch Lecture |
| Integral Test Intuition | Khan Academy | Use this as a quick visual introduction if the theorem feels abstract. It emphasizes the rectangle-versus-area picture, which can make it easier to remember why a decreasing positive function can be used to test a related series. | Watch Primer |