Baby’s First Infinite Series


Okay, okay, I admit this isn’t suitable material for babies, nor for any normal kid whose age can be expressed with one digit! The “baby” phrase just popped into my head and I found it too cute to resist. A more accurate title might be “Joanie & Johnny’s First Infinite Series”, or “Kids’ First Infinite Series”.

I’m always thinking about ways to get those apathetic middle- and high-school students interested in math again by showing them something wild and crazy, yet simple and concrete enough (for them, of course) that it shouldn’t be too difficult to understand. My latest idea is an infinite series with a surprising and very simple visual “proof”.  Can you guess what it is? Hint:  the “proof” is in each of the figures above (surprise, surprise). What it proves is…


I’m sure each of these informal proofs — really variations on a single proof  — has appeared in many places, but I ran across the one on the left first, in Roger Nelsen’s wonderful Proofs Without Words. (The one on the right is from the Wikipedia article “Geometric series”.) The two volumes of Proofs Without Words, published by MAA, contain dozens of marvelous and miraculous visual “proofs” on a wide variety of mathematical topics; I highly recommend browsing through them. Full disclosure, though: in my opinion, not many are as elegant or as easy to see as this one. Still, I believe that several of the proofs Nelsen has collected, certainly including this one, could be presented successfully on the middle-school level — though probably not with the above notation. Regardless of notation, of course, appreciating this proof requires some ability to do arithmetic with fractions.

I think most people with experience teaching high-school math would agree that (1) not many “apathetic” secondary-school students have much of an idea what a proof is (notwithstanding the emphasis on them in a typical geometry course), and (2) hardly any of them know why they or anyone should care! That may be a problem with my idea, but it strikes me more as an opportunity. I suspect that going through something like this proof would help with both problems; it would also expose students to the important idea of adding up infinitely many numbers and getting a finite sum.

Do you agree? Do you this could work (with a reasonable amount of scaffolding, naturally) as a discovery lesson for, say, 10th graders? If you think it’s worth trying, here’s an idea for “reasonable scaffolding”: Discuss the fact that 1/3 = 0.33333333…, with infinitely many 3’s, and ask the class what 0.33333333… with infinitely many 3’s actually means. Of course it’s a very simple infinite series, one with the same sum as the series I’ve been discussing:

InfiniteSumFormula2Since both series have the same sum, an obvious question might be whether it’s possible to draw a picture to illustrate the above equation, and if so, what it would look like. I leave those questions as an exercise for the reader.


Zeno’s “Achilles and the Tortoise” Paradox and Geometric Series

When I decided to retrain as a secondary-school math teacher and applied for a Woodrow Wilson Indiana Teaching Fellowship a few years ago, I had to teach a 5-minute lesson. It could be on any subject I wanted, and I chose Zeno’s first paradox of motion, the “Achilles and the Tortoise” Paradox. As you may know, the Greek philosopher Zeno (5th century BCE) devised several “paradoxes of motion” that baffled all of his contemporaries. His first paradox involves a race between Achilles (a very fast runner, said to be “the fleetest of foot of all mortals”) and a turtle (a rather slow crawler), with the latter having a head start. Zeno gives a simple but surprisingly convincing argument that Achilles would never catch up: “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

In my brief lesson, I explained that this argument assumes that the sum of an infinite geometric series cannot be finite, and showed why that’s not true, i.e., why such a series can converge to a finite value.

Why, in my interview lesson, did I teach something as advanced as convergence of infinite geometric series? In two words, intrinsic motivation. High-school Algebra II (at least in Indiana) covers geometric series and their sums, and infinite geometric series are very likely to be touched on. Still, students won’t be expected to understand why the sum of infinitely many numbers can be finite until they take calculus. But, while the details require methods of calculus, the concepts are simple; there’s no reason they can’t understand immediately. I’m fond of the infinite, that fruitful source of crazy ideas, and I’ve wanted from the beginning to expose students to the wildest and craziest ideas I could possibly get them to understand. And why did I want to do that? The main reason is that, based on my own experience, I believe that wild and crazy ideas can be a great source of intrinsic motivation for a lot of kids—certainly not the majority, but a lot. And intrinsic motivation is generally agreed to be far more reliable than external motivation. Along the same lines, Sawyer’s fine book Prelude to Mathematics comments “A non-mathematician learning mathematics…often has to plough through routine procedures, which can be extremely dull… An education should also contain elements that perform the functions of a cold bath—to provide a shock and keep one awake.”

I’ve given versions of this lesson three times now, including once to a high-school “math exploration” class, and I think it went over well each time. The latest version — available in narrative form — has two parts: “Resolving the Paradox” (roughly the original 5-minute form) and “Boring Details, A Related Example, and the General Situation” (about infinite geometric series in general and when they converge). The only technology needed is a yardstick and a greenboard drawing.