How to teach sums and asymptotics to students who don’t know limits

I am to teach sums and asymptotics to first year CS undergraduates who have only high school math. I thought it would be attractive to follow chapter 14 of https://courses.csail.mit.edu/6.042/spring18/mcs.pdf as far as possible.

However the students will have never seen limits before. What is the minimum amount of material I need to give a coherent picture of how to understand and compute infinite sums? I would like the course to be self contained as far as possible.

• Welcome to Computer Science Educators! This is a very interesting question. Could you perhaps explain, in more details, what mathematical background the students do have? – ItamarG3 May 12 '18 at 13:34

For asymptotes in general, one of the easiest examples for beginners is the curve y = 1/x which has asymptotes (at infinity) on both axes. It is easy for them to see that as x gets large y gets small (without bound). Likewise as x gets small, y must get large. You could then look at y = 3 + 1/x for example to see a different horizontal asymptote. Contrast this with y = x^2 for an unbounded function and also with a periodic function (sin or something informal that looks like it), that is bounded but has no asymptote.

If you want to go farther, you could discuss a chord and a tangent on a circle. Move the chord parallel to itself and see the two intersection points with the circle converge until you have a tangent parallel to the original chord. This may be more than you want or need, of course.

For infinite series and sums, one of the easiest examples is to first discuss Zeno's Paradox and the sequence 1/2, 1/4, 1/8, ... and its sum. It is pretty clear if you draw the picture that the sum is one and that the difference between a partial sum and 1 is the most recent term in the series. It is also easy to see how to generate the next term.

If you are teaching programming, you can write a program that computes partial sums, but if you do so with floating point then you also need to discuss rounding error, which will get severe, since the sum gets larger as the terms get smaller. Eventually, then, the next term adds nothing at all, due to the fixed precision of floating point data.

But if you want to compute the partial sum in closed form then you can do better in a program to compute that rather than the sum itself. That may be too much to ask, I realize.

That may be enough, but it gives enough of the overall concepts to get you started. And Zeno is always fun in any case.

Here are a few more "interesting" functions that may go beyond what you need, but have interesting things to teach you.

Once you understand what f(x) = 1/x does, you can combine it with other functions to get other sorts of asymptotes. For example g(x) = x + 1/x has is asymptotic to the function h(x) = x. In that form it is pretty obvious what it does, but less so if you express it as g(x) = $(x^2 + 1)/x$.

Other combinations are possible. If you look at k(x) = $(1/x)*sin(x)$ you get a function that oscillates but is also asymptotic to zero at infinity.

One of my favorite functions is m(x) = $(1/x)*sin(1/x)$. As x approaches infinity, this is asymptotic to zero also, but it is much more interesting as x approaches zero. It has the property that it takes on every real value infinitely many times in every neighborhood of zero. The 1/x factor gives it increasing amplitude near zero and the argument to sin makes it oscillate faster and faster. This is a very strange property for a real valued function, but if the student will eventually study Complex Analysis it is an exceedingly important example. I'll leave that explanation for the next course.

I mean...limits aren't too bad. You could always just explain limits too.

Limits, intuitively, are a way to "approach" a number when you can't actually reach it because your expression would be undefined. For example:

$$\lim_\limits{x\rightarrow 0} \frac{1}{x}$$

One cannot just plug 0 in for $x$ - the answer is undefined. So we must "approach" the solution using the limit.

There are several different methods for solving limits. Most are detailed in this great series of YouTube videos. (I also have some old notes on the subject here; obligatory "these are old and out of date" warning.) I could imagine giving this intuitive explanation, showing a few examples of what it means, watching the videos with the class (or explaining the points in your own way) and doing a bunch of practice problems, and then the students being fine.

You could also, of course, show them the epsilon-delta formalization if you feel it's necessary.

• I am intrigued how notes on limits can be out of date :) – Anush Aug 3 '18 at 17:14
• @Anush lol, fair point. I suppose I mean, they are in probably very few ways the way I'd write those notes now, and they probably have a few errors, as I wrote them while learning about limits (though rewriting them now would require as a refresher, as I cannot remember the last time I actually calculated a limit). – heather Aug 3 '18 at 17:37