# Emphasizing the decreasing condition in the Integral Test or in the AST (in Calculus II): is it worth the time?

The title is basically the question. But I guess I should expand a little. For the background, I'm teaching at a large public university in the US. Out student body is mixed in terms of their abilities, but most students aren't strong, to say the least (and it's been going downhill lately). Calculus II is one of the courses that I frequently teach.

And now to the question: for a while I've been wondering if I'm wasting class time when working on the condition that the function in the Integral Test for series has to be decreasing. Let me explain.

As we all know, the Integral Test for series requires that the function $$f$$ be positive, continuous, and decreasing on the interval $$[1,+\infty)$$ (or starting wherever). When teaching the test, first we do examples like $$\sum\frac{1}{n^2}$$ or $$\sum\frac{1}{\sqrt{n}}$$ to lead to the $$p$$-series. Then we'll do some more examples like $$\sum\frac{17}{n\ln(n)}$$. What all these examples have in common is that the "related" function $$f$$, such as $$f(x)=\frac{1}{x\ln(x)}$$ in the last example, is obviously decreasing. But then there's an example like $$\sum\frac{\ln(n)}{n}$$ … and I'm having some trouble with examples like these.

• On the one hand, of course, for a rigorous and a fully correct solution, we need to show that the function $$f(x)=\frac{\ln(x)}{x}$$ is decreasing, at least starting somewhere. And there are several teachable moments here! One is to demonstrate the importance of a theorem, such as a test for series, being an "if-then" statement; in other words, it's a nice lesson in logic and rigor of math. Second, every time when I ask if this is decreasing, I always get the response that "Yes, because it goes to zero", and so this prompts a discussion of how going to zero is not the same as monotone decreasing.
• But this is precisely the problem — this discussion takes time! And so does taking the derivative to demonstrate that the function is decreasing, or to find the interval where it indeed is. And where I teach, considering the students I teach, time is extremely precious. I simply don't have enough time to do everything that I'm supposed to do plus the things that I'm not supposed to do but still have to (like repeatedly explaining to my Calculus students that $$\sqrt{x^2+1}\neq x+1$$, $$\frac{1/3}{2}\neq\frac{2}{3}$$, and the like — because that's the kind of students we have in our Calculus classes).

So I'm feeling more and more inclined to kinda ignore this decreasing requirement in the Integral Test (and the Alternating Series Test too). By "kinda" I mean that, of course, I'm still going to state the correct theorems, but not do any examples where checking the decreasing behavior is necessary.

Any thought on this issue? I'd like to pick this community's mind on this.

• "but not do any examples where checking the decreasing behavior is necessary." That's quite a strange wording in my opinion. Since the condition is part of the theorem, it is always necessary to check it when one applies the theorem. "Examples where the assumption is satisfied" is not the same as "examples where one does not need to check the assumption". Commented Aug 11 at 6:20
• I cut my lengthy response, which was basically just saying "do less, but do it well". You could also look at this similar question from the past: matheducators.stackexchange.com/questions/13912/… Including the past arguments for less/better versus more/worse. Commented Aug 11 at 15:36
• How is this covered in the textbook? Commented Aug 12 at 4:19
• @JochenGlueck : I can see that my wording was a bit confusing. I meant to say that it's examples where checking the decreasing condition doesn't require any work. I still insist that my student go thru the checklist of the three conditions. But with series like $\sum \frac{1}{n}$, the answer to "Is the function $f(x) = \frac{1}{x}$ decreasing on $[1,+\infty)$?" is obviously "yes" without any work. Commented Aug 12 at 20:29
• @DanielR.Collins : With both kinds of examples. I mean, the book has both examples like $\sum \frac{1}{n}$, where decreasing is obvious, and like $\sum \frac{\ln(n)}{n}$, where we need to differentiate the function. But the book (being a typical US Calculus textbook) is way too long anyway, so I don't feel pressed to do the impossible and cover everything in the book. Commented Aug 12 at 20:54

You don't need too much time to emphasize the importance of the decreasing condition. All you need to do for that is

a) To produce a simple example where the integral test doesn't work. If you have hard time inventing one yourself, I suggest $$f(x)=\sin^2(\pi x)$$ (the series $$\sum_{n=1}^\infty f(n)$$ converges but the integral $$\int^\infty f(x)\,dx$$ diverges) and $$f(x)=\frac 1{1+x^6\sin^2(\pi x)}$$ (the other way around). That's for those who love formulas. Pictures are obvious and I would present them too. Should take about 25 minutes if you know what you are doing yourself.

Note that all "normal" functions we give the students (like the ones you mentioned) are in some Hardy field, so they are eventually monotone and sign preserving, hence you are safe with them. However, explaining that to calculus 2 students will, indeed, take too much time though you may state the corresponding general theorem without proof and allow the students to quote it.

b) To reduce a noticeable amount of points every time the student forgets to say that the function in question is decreasing. You don't need to ask them to prove it every time, but they still have to know that this is a necessary condition. Takes 0 time but some sternness and determination.

c) To make a handout explaining the full proof of the integral test with clear examples showing the necessity of every assumption. The preparation of such handout may take a couple of hours, but the distribution should not take more than 5 minutes (you should also announce that questions about the handout are welcome during the office hours).

These three simple steps will allow the ones who are capable of learning to understand everything, and, to be frank, you have no chance to teach the remaining ones anything anyway, so don't even bother to try.

Just my two cents, as usual :-)

• Krantz in How to Teach Mathematics makes the point that handouts are almost always a waste; they exist because an instructor feels guilty about not covering something properly in class, and are almost always ignored by students. Commented Aug 12 at 4:18
• @DanielR.Collins Steven Kranz makes many points I disagree with in many of his books. I use handouts quite a lot from supplementary material to samples of what I expect as a proof on the exam (even if the theorem was thoroughly covered in class) and they are usually well-received. I have never "felt guilty" when giving them and as to "ignoring them", it is student's choice: they may opt to ignore the entire class and I consider it their problem: in that case if they can still pass the exams, fine with me, and if they fail, I do not cry for long :-) Commented Aug 12 at 10:11
• "Just my two cents, as usual :-)" ← Looks to me more like your three cents. :) Commented Aug 12 at 20:58
• So, about your three cents, I mean, your three points. But first of all, thank you very much for a wonderful answer! (a) "Should take about 25 minutes" kills this right away. The whole reason I asked this question is because I don't have enough time. That's not to say that I don't like it, because I actually do! But I most certainly don't have the time for this. (b) As I clarified in a comment under my original post, I always do that. It was a fault with my wording that I didn't make that clear. Sorry… (c) This is a great suggestion! And moreover, it incorporates your own suggestion (a). Commented Aug 12 at 21:16

It seems like you're laboring under the misconception that every detail of every topic has to be covered orally during class time. There is a reason that you have a textbook.

I would certainly not spend time discussing this in class, because it's basically never relevant when people are using the test the way they really use it. In real life, you're typically taking the closed-form expression for the series and extending that expression to the whole real line. Since the original integer expression was decreasing (by hypothesis), typically so is its extension to the real line.

• "It seems like you're laboring under the misconception that every detail of every topic has to be covered orally during class time." No, I don't. That I'm struggling with one topic doesn't imply that I have the same issue with all the topics. (Sorry, but your logic reminds me of the "proof" that all odd numbers are prime from the observation that .3, 5, and 7 are.) Commented Aug 12 at 21:21