# What strategy for picking convergence tests for series do you teach?

Without getting bogged down in details, I'll list the names only. It seems that the strategy I generally use is this:

1. Divergence test first

2. Is it a recognizable form? p-series or geometric?

3. a) No negative terms? Integral, direct comparison, limit comparison?
b) Possibly negative terms? Alternating series test, root test, ratio test?

Those are my main categories, and then each test in the category has its own favourite type of series to work with (like ratio test with factorials and nth powers, for example) (edit: btw, the tests in each group aren't in any particular order)

But oddly, the most common thing I stress when teaching series is perseverance and practice since it's often about recognizing known patterns in unknown series

• Why is such a strategy important in the first place? – Peter Saveliev Aug 6 '20 at 16:02
• I think everyone uses a strategy to some degree. For example, would $\sum 1$ converge or diverge? Obviously not, since you're always adding another number. Additionally, I don't see why one would want to use a test that only works for positive terms for series that contains negative terms. It seems like without some basic strategy, students would pick at random. – Robbie_P Aug 6 '20 at 16:20
• I basically pose the same order to my students, except I tend to favor direct comparison well-before the integral test. – Nick C Aug 6 '20 at 16:28
• @NickC actually, I just randomly listed them once I had each one in a group (but thanks for the comment since it clarifies things) – Robbie_P Aug 6 '20 at 16:31
• @NickC Often the integral test is done before the comparison test because you would like to establish the result on p-series (so you have more to compare to). – Steven Gubkin Aug 6 '20 at 18:52

I made this flowchart for my students last time I taught this stuff. Not the best visually, but I think it effectively conveys my thought process.

• Nice chart. Can you share how you created it (what software)? – Mark Fantini Aug 6 '20 at 21:54
• To your Note at the bottom-left, do you cover any series that are built from geometric series, such as the sum of n(1/2)^n. Not sure how common that is in calculus before, say, combinatorics. – Nick C Aug 7 '20 at 0:13
• @MarkFantini I think I used an online flowchart app. – Steven Gubkin Aug 7 '20 at 0:36
• @NickC When we do power series, I include some problems which guide the students towards evaluating such sums (in this case, using term by term differentiation). It isn't in the curriculum for them to learn these tricks though. – Steven Gubkin Aug 7 '20 at 0:41

I think this is one of those places where teaching a detailed strategy is a form of "teaching to the test" that is counterproductive for the students' intellectual development.

It's important to emphasize that all the convergence tests have preconditions that must be satisfied before one can use the rule, and one should not use a test without explicitly checking that those preconditions are satisfied.

As students advance in their studies (no matter what they study), and when students use their knowledge outside of the classroom, they will often be in situations where they have a problem that they haven't been taught how to solve, but they do know a number of different potential strategies for solving the problem and will have to pick one or more to try. The problem of testing a series for convergence is a good problem on which to develop the general skill of approaching and solving this kind of problem. Part of the point of the course is to give them practice, with guidance, at developing their own strategy for this kind of problem.

Sine finding formulas for antiderivatives is a similar kind of problem, I tell students on the first day of Calc II that the main purpose of the course is to develop this general skill, and in fact that this main purpose is more important than any kind of problem they learn how to solve (which, after all, can all be done by our favorite symbolic computation software).

• (1/2) I too feel that it does feel like training "computers" rather than students, but this topic is basic enough to be in an 1st year calc course. Thus, are learning strategies for finding tests for convergence any more than more teaching to the test than strategies for differentiation? For example, "If you have a product of functions $f$ and $g$, then product rule, but if you have a quotient of them, then it's quotient rule." If that's strategic or algorithmic, would that also be counterproductive? If the work could be done by a computer then it might be inescapably algorithmic – Robbie_P Aug 7 '20 at 6:02
• (2/2) I suppose my approach would be to get through the strategy (based on preconditions) quickly so that we can get to more interesting examples that aren't just computation. So I would, rather than have classes where we do problems where a student is mimicking what a computer could be programmed to do, the student could do problems that a computer couldn't do - like proofs, word problems. – Robbie_P Aug 7 '20 at 6:10
• The difference between differentiation and testing for convergence is that, in the case of differentiation, there is usually only one rule that is legal to apply. In the case of testing for convergence, there are frequently several tests that are legal to apply, but some of the legal tests will not be useful to apply. – Alexander Woo Aug 7 '20 at 6:15
• When it comes to testing for convergence, what humans do is different from what computers do. A computer has lots of memory and processing power, so it is programmed to simply try all the legal tests and throw away the ones that give inconclusive results. A human on the other hand tries to figure out which of the legal tests is useful before even trying it, and might abort a legal test part way through if it looks like it won't be useful, perhaps returning to it if other legal tests seem even less likely to be useful. – Alexander Woo Aug 7 '20 at 6:18
• Hmm, I didn't know that computer would do that but it seems extremely inefficient. Like I would have expected something like "If 1 is true then try 2 based on a few sample computations and see if there is a geometric ratio or somethign. If 2 is false, then are there negative terms? If negative terms is true, then 3b, else 3a." – Robbie_P Aug 7 '20 at 7:02

I don't teach this course, and I think the desire to have an algorithm this detailed may be an example of the kind of thing that people will start to feel because they're teaching the class but that people in the real world don't actually use. Students want to be told that everything is rule-based, because it makes them feel safer.

When I encounter this kind of thing in a real-world context like physics, I basically reason by analogy, and that pretty much always works.

For example, I know that $$\sum n^{-1}$$ diverges, but is on the ragged edge of convergence. This tells me that an example like $$\sum 1/n\ln n$$ may or may not converge (the extra log factor might tip it over the edge). However, if it does converge, it will converge so slowly that it would be useless to try to evaluate in any practical context. So in an application, I'm done. I don't pursue this example any further.

I don't think people in the real world know a list of tests, and in particular they don't think in terms of an "integral test." They think in terms of an integral analogy. If someone asks me to explain how I know that $$\sum n^{-1}$$ diverges, I'll say, "Oh, it diverges logarithmically." Never mind that what produces a log is the integral, not the sum -- everybody knows what is meant by this.

If for some reason I really wanted to know for sure whether $$\sum 1/n\ln n$$ converged in theory, the first thing I would do would be to try it in a CAS and see what it said. If it said it diverged, I would believe it. If it said it evaluated to some exact expression, I would also believe it. If it spit out a decimal approximation, I would know not to trust it. When I did this in the CAS Maxima, for this example, it said it didn't know.

OK, so if I really still cared, the next thing I would do would be to try the analogous integral, which in Maxima is integrate(1/(x*log(x)),x,1,inf);. It tells me it diverges. OK, cool, I'm done if all I want to know is the answer.

In the real-world context I probably don't care about an actual proof, but if I did, I'd see if the CAS could do the indefinite integral. It turns out that it can, and the result is $$\ln\ln x$$. This shows me that it diverges, but ever so gently. If I care about writing up a human-readable argument to this effect that doesn't depend on sofwtare, then I can check that the derivative of $$\ln\ln x$$ really is $$1/x\ln x$$.

• So it seems like in applications and practice, the answer is "rather than a student do the work, have a computer do the work"? But how would a student do a question without a CAS or, if they did, then how would they know the computer did it correctly? I don't know how computational programs work, but it seems plausible that it would have been programmed in a logical way, and so wouldn't the algorithmic approach still apply (even though it's a computer doing it and not a student)? – Robbie_P Aug 7 '20 at 3:16
• @Robbie_P_math: So it seems like in applications and practice, the answer is "rather than a student do the work, have a computer do the work"? No, the part of my fictional story involving software is a part that in reality I would never get to. I would stop after observing that, based on analogy, there's simply no way this series is going to be useful. But how would a student do a question without a CAS By analogy. or, if they did, then how would they know the computer did it correctly? I describe at the end of my answer what I would do. – Ben Crowell Aug 7 '20 at 14:21
• @Robbie_P_math: I don't know how computational programs work, but it seems plausible that it would have been programmed in a logical way, and so wouldn't the algorithmic approach still apply (even though it's a computer doing it and not a student)? Humans aren't computers. Computers play chess by brute-force computation, while humans play by pattern recognition plus a very small amount of computation. Computers do integrals using algorithms like the Risch algorithm, but humans do them by pattern recognition. – Ben Crowell Aug 7 '20 at 14:23
• I agree that this is generally how I would think about convergence if I ran across a problem like that in the wild, but I can’t help but wonder if my intuition and analogy come from a place of having understood all those convergence tests at one time (even if I couldn’t rattle them all off the cuff for you now without a bit of hesitation). – Eric Aug 8 '20 at 7:52

First, let's ignore geometric series and $$p$$-series because those are standard examples.

For infinite series with positive terms, if you really understand how sequences grow then almost all examples that occur in basic courses can be handled by the limit comparison test unless the sequence has factorials in it, in which case you use the ratio test.

The root test is largely irrelevant in basic courses because no important series with positive terms requires it. A student who goes on to study real math will find out that in fact the root test is theoretically very important as the basic idea behind Hadamard's radius of convergence formula (using $$\varlimsup$$ instead of $$\lim$$).

I taught this subject only once (two years ago). In that occasion, I wrote an one page summary for my students in the following order.

1. Divergence test
2. a) Geometric
b) p-series
c) Alternating
3. Ratio
4. Root
5. Integral
6. Comparison