One common question from students first learning about series is how to know which convergence test to use with a given series. The first answer is: practice, practice, practice. The second answer is that there is often more than one convergence test that can be used with a given series.
I will illustrate this point with a look at one series and the several tests that may be used to show it converges. This will serve as a review of some of the tests and how to use them. For a list of convergence tests that are required for the AP Calculus BC exam click here.
To be able to use these tests the students must know the hypotheses of each test and check that they are met for the series in question. On multiple-choice questions students do not need to how their work, but on free-response questions (such as checking the endpoints of the interval of convergence of a Taylor series) they should state them and say that the series meets them.
For our example we will look at the series
Spoiler: Except for the first two tests to be considered, the other tests are far more work than is necessary for this series. The point is to show that several tests may be use for a given series, and to practice the other tests.
The Geometric Series Test is the obvious test to use here, since this is a geometric series. The common ratio is (–1/3) and since this is between –1 and 1 the series will converge.
The Alternating Series Test (the Leibniz Test) may be used as well. The series alternates signs, is decreasing in absolute value, and the limit of the nth term as n approaches infinity is 0, therefore the series converges.
The Ratio Test is used extensively with power series to find the radius of convergence, but it may be used to determine convergence as well. To use the test, we find
Since the limit is less than 1, we conclude the series converges.
A series, , is absolutely convergent if, and only if, the series converges. In other words, if you make all the terms positive, and that series converges, then the original series also converges. If a series is absolutely convergent, then it is convergent. (A series that converges but is not absolutely convergent is said to be conditionally convergent.)
The advantage of going for absolute convergence is that we do not have to deal with the negative terms; this allows us to use other tests.
Applied to our example, if the series converges, then our series will converge absolutely and converge.
The Geometric Series Test can be used again as above.
The Integral Test says if the improper integral converges, then our original series will converge absolutely.
since ln(1/3) < 0.
The limit is finite, so our series converges absolutely, and therefore converges.
The Direct Comparison Test may also be used. We need to find a positive convergent series whose terms are term-by-term greater than the terms of our series. The geometric series meets these two requirements. Therefore, the original series converges absolutely and converges.
The Limit Comparison Test is another possibility. Here we need a positive series that converges; we can use again. We look at
and since the series in the denominator converges, our series converges absolutely.
So, for this example all the convergences that may be tested on the AP Calculus BC exam may be used with the single exception of the p-series Test which cannot be used with this series.
While the convergence of the series used here can be done all these ways, other series lend themselves to only one. Stress the form of the series that works with each test. For example, the Limit Comparison Test is most often used for rational expressions with the numerator of lower degree than the denominator and for expressions involving radicals of polynomials. The comparison is made with a p-series of whatever degree will make the numerator and denominator the same degree allowing the limit to be found.
Most textbooks, after explaining each test and giving exercises on them, include a series of mixed exercises that require all the test covered up to that point. A good way to use this set is to assign students to state which test they would try first on each series. Discuss the opinions of the class and work any questions that students are unsure of or on which several ways are suggested.
Of course, I didn’t really answer the question, did I? Check next Friday’s post and I’ll try again.