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.


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