AP Calc 10.6 Comparison Tests for Divergence

Note: I’ll be using an and bn to denote each series since I can’t do subscripts.

Direct Comparison Test

The comparison test is a method to determine whether a series converges or diverges. The definition of the test is as follows:

In other words:

If you have a convergent series whose terms are always larger than the terms of the original series, the original series must be convergent.
If you have a divergent series whose terms are always smaller than the terms of the original series, the original series must be divergent.

The test does not apply when the terms of the divergent series are always larger or the terms of the convergent series are always smaller. Be careful with this! This test will not always work. It requires either a known larger convergent series or a known smaller divergent series.

Examples

Worked Solutions

Solution a)

Solution b)

Solution c)

Limit Comparison Test

A useful way to use the limit comparison test is to compare a complicated algebraic series with a p-series. We’ll look at some examples below:

Examples

Hint: Two can also be solved with the direct comparison test.

Worked Solutions

Solution a)

Solution b)

Solution c)

Solution d)

You could also use direct comparison with the divergent series ∑1/n since ∑1/√n is not smaller than the original series.

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. The first tab has notes explaining the tests + proofs for those interested. The other two tabs are dedicated to 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.