# Calculus Teaching: Is it possible or desirable to give a severely abbreviated treatment of series convergence tests?

I will be teaching Calculus 2 this fall at a large U.S. state university. Our incoming students tend to have a limited or inconsistent background, which limits the amount of material we can cover.

Previously at my university, instructors have given sequences and series a very thorough treatment, including all the usual tests for convergence (ratio, root, alternating series, comparison, integral, etc.). For lack of time, important subjects such as topics such as arc length, parametric equations, polar coordinates, etc. have had to be dropped or abbreviated. I want to cover these subjects, and therefore am looking for something else to cut.

It seems possible to give an abbreviated treatment of sequences and series: cover the basics, do only the Ratio Test, and then teach Taylor and Maclaurin series. The students will be able to determine the radius of convergence, although perhaps not the endpoints.

Have people tried this approach previously? And are there disadvantages to such an approach, apart from the ones which can be easily foreseen?

Thanks.

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Did the "Community Wiki" option disappear? I would have clicked the button, but it no longer seems to exist. – Frank Thorne Jul 12 '13 at 13:58
It seems that there are a small number of big ideas: (1) The geometric series can be explicitly summed; (2) Whether or not a series converges is a property of its tail; (3) If the tail of a series which is known to converge/diverge (in practice the geometric series) dominates/(is dominated by) the tail of some other series, then that series necessarily converges/diverges. It seems to me that if you can get these points across you will have done your students a great service. Caveat: I have no experience trying this in practice and am not a scholar of education. – Aaron Hoffman Jul 12 '13 at 16:02
I think the main issue would be whether subsequent courses have your course down as a prerequisite where they are supposed to learn "sequences and series". If not, then fine; if so, then probably consultation with the lecturers of those other courses is in order – Yemon Choi Jul 12 '13 at 18:30
I had a meeting yesterday with some engineering faculty at my school to discuss what parts of the first-year calculus curriculum are useful to their 0undergraduate students, and they (the faculty) had no interest in all those convergence tests. They'd much rather students see Fourier series than spend 2 weeks (?) on bazillion convergence tests for power series. – KConrad Jul 12 '13 at 22:05
Out of curiosity because I do not know the system well: Am I right that skipping most of the proofs is normal in US calculus lectures? – The User Jul 12 '13 at 23:14

The Harvard Consortium text

Calculus: Single Variable, Course Advantage Edition, Third Edition by Deborah Hughes-Hallett, Andrew M. Gleason and William G. McCallum

http://bcs.wiley.com/he-bcs/Books?action=resource&bcsId=1404&itemId=0471448761&resourceId=3360

gives a very perfunctory treatment of convergence of sequences (almost none in the first edition) -- which is somewhat ironic given its heavy reliance on numerical methods. On the other hand, it manages to treat parametric equations, polar coordinates and even a few tests for convergence of series, besides other desirable topics (Taylor series included). I taught from it (well, from the 1st and 2nd edition, to be precise) several sections of Calculus I and II at a "big state university" a few years ago, and while I was not happy about the irony, I must admit the book is quite efficient and generally not boring-- unlike another calculus book from which I taught later at another "big state university".

To whoever might read this: I am just describing my actual teaching experience, not taking a stand of any kind in the debate on "reform calculus". This textbook (just like any other) has its pros and cons, many of which are summarized here:

http://www.math.harvard.edu/~knill/pedagogy/harvardcalculus/

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I've taught a treatment of series that was similar to what you are describing, if not quite as abbreviated. The primary difficulty I would point out is that you need the root test for a "good" (constructive) proof that Taylor series have a radius of convergence. There's a correct but completely non-constructive proof using I think the comparison test. If you actually want to compute the radius of convergence in real examples you of course use the ratio test; the problem with using this for a proof is that it fails if the series has terms equal to zero.

I'd also point out that if you want to prove that the ratio test works, you need some of the other tests. (E.g., if I recall correctly, this can be proved by showing that the geometric series converges for ratio less than one, and then using the comparison test.)

Nevertheless, if you're willing to handwave a lot of stuff, something like this can be made to work. My only other comment is that you should be wary of trying to rush through Taylor and Maclaurin series: a lot of students have trouble with these, and they are quite important in subjects where mathematics might be applied--much more so, I believe, than convergence tests, polar coordinates, or arc length.

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Thanks Charles. The level of our students is such that I feel I should skip most or all of the proofs. Especially since there is a separate honors section of calculus. – Frank Thorne Jul 12 '13 at 16:12
Actually, I was in a similar situation--the course I was teaching was specifically targeted at students with poor precalculus backgrounds. However, at my university, which is known for being very theoretical in all respects, proofs are heavily emphasized even at this level. So, for instance, by this point in the course, my students more or less understood how to do a very (very!) basic epsilon-delta proof even if they could not add fractions. – Charles Staats Jul 12 '13 at 19:26
I would note that I think it's important to be able to address why the series always diverges outside the radius of convergence, since this can be counterintuitive in many examples (e.g., when $f(x) = 1/(1-x)$, it's intuitive that the series diverges for $x \ge 1$ but not for $x < -1$). But this example can be done using the ratio test, along with some rough explanation that the divergence is necessary because the summands keep getting bigger rather than smaller. – Charles Staats Jul 12 '13 at 19:31
Oh--one other note: I can attest from my experience that basically skipping sequences and going straight to series does seem to work. – Charles Staats Jul 12 '13 at 19:33

Is it possible? Of course, it is, as you have implied. Is it desirable? It much depends on your big picture of the course and the bigger picture of the previous and the next courses. Any way, here is what I have done in rather "similar" courses where one should skip proofs. Strangely, I start with Maclaurin Series! For more details see here. Since, you are not going to give proofs, you should replace them with some convincing arguments. And, the "convergent graphs" work well in that direction. Moreover, unlike what you have suggested, discussing the end points would be very useful for bringing about some simple and intuitive cases of the convergent tests. In particular, "alternating series test" would appear early on and you may provide an intuitive and pictorial argument for that. Generally speaking, for such a course as you described, it is not that much important whether or not you cover this test or that one, It would be much more important to give your students a sense of what such tests are designed for, and how they work. Thus, the ratio test (as you suggested above) and the alternating series tests would suffice. They may learn the rest when they need them.

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