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

Question

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.

Answer 1

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 :-)

Answer 2

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.