The Integral Test
The integral test compares a series to its integral to prove convergence or divergence!
This test is best illustrated by examples, so let’s look at some of them together.
Examples
Worked Solutions
NOTE: I forgot the dn in the first two integrals, don’t make my mistake!
We can apply the integral test since f(n) is continuous, positive, and decreasing between 0 and infinity.
Since the integral diverges, the sum diverges.
We can apply the integral test since f(n) is continuous, positive, and decreasing from 1 to infinity.
Since the integral converges, the sum also converges.
Apart from knowing how to integrate, the key point in applying this test is knowing the conditions it requires. Remember:
To apply the integral test, f(n) must be
- Continuous
- Positive
- Decreasing
On the interval the series is asking for (usually 1 to infinity).
Flipped Math
Flipped Math is an awesome resource with full packets, test prep, and corrective assignments for each lesson!
Math Libretexts
Math LibreTexts is a free online resource packed with textbooks and learning materials for math students. It simplifies complex concepts, provides practice examples, and offers interactive features for convenient learning anytime, anywhere. It’s like a superhero ally for math students, making math easier to understand and excel in.
Paul's Online Notes
Paul’s Online Notes is another amazing resource with in-depth explanations and a few examples. It has three tabs, one with notes and the other two with practice problems.
Khan Academy
A classic resource for a reason, Khan Academy offers a ton of practice resources with explanatory videos. However, they’re known for being easier than the actual AP test so you might want to supplement with other resources to make sure you’re prepared for test day.