Most students feel more comfortable with the Limit Comparison test than Direct Comparison since they test limits with infinity frequently. So, is it imperative to learn the Direct Comparison test or will Limit Comparison solve all of the same series convergence questions?
Short answer: for the most part, the Limit Comparison test will work on any series convergence question that a Direct Comparison test will work. See the example below, determining if the given series converges or diverges.
It’s worth noting that although the Direct Comparison test is another convergence test to know/memorize, it is significantly less work to determine convergence or divergence than its counterpart, the Limit Comparison test.
How to use the Direct Comparison Test
Without jargon, to use the Direct Comparison test, we first identify a comparison series (typically a geometric or p-series that our given series looks similar to). Next, determine whether that comparison series converges or diverges.
1. If the comparison series converges, we need the terms of our given series to be smaller than the terms of the comparison series to also converge.
2. If the comparison series diverges, we need the terms of our given series to be larger than the terms of the comparison to also diverge.
Showing that the terms of our given series are larger or smaller than the terms of a comparison series can be challenging for students. A helpful hint for a lot of problems, since they're usually fractions, is to compare denominators. The larger the denominator, the smaller the number, as seen in the worked out example above. Or, in the second example below, the smaller the denominator leads to a bigger number.
As you can see above, there are some series where the Limit Comparison test fails, but Direct Comparison test works.
However, for this example you could use the Integral test to get the same conclusion of divergence. In order to use the integral test, it requires a review of which integration technique to use.
For other series, the Ratio test or other convergence tests can be used to get around the direct comparison test.
So far, the Direct Comparison test isn't imperative to know.
The one time you must know the Direct Comparison test is for a theoretical multiple-choice question like FlippedMath created below:
These theoretical questions appear on the AP Calculus BC exam, so they’re worth knowing and practicing. This question is created from the conditions needed to use the Direct Comparison Test. So even though you don’t have to apply the Direct Comparison Test to solve the problem, you still need to understand how Direct Comparison can be used.
The answer to the problem above is B. For additional practice problems and solutions, check out the 7 best resources to study for AP Calculus tests. You'll recognize the second link is Flipped Math, where I found the last example problem.
Here are a few more worked out examples using the Direct Comparison Test.
As you can see in this last question, the Direct Comparison test is important to know and apply.
At the end of the series convergence unit, you should know the following convergence tests:
nth term
Geometric
Telescoping
Integral
p-series
Direct Comparison
Limit Comparison
Alternating Series
Ratio
Root (optional, not needed for the AP Calculus BC exam)
If you need more help determining the appropriate series convergence test to use, or applying the various convergence tests, join me for tutoring!