This is the first guest post from John Chase’s dad, also a math teacher. Thanks, son, for letting me post to your blog.

Gene Chase: I was taking a shower today when I figured out why I always confused the words “sequence” and “series.” 2, 3, 4, 5, … is a sequence; 2+3+4+5 is a series. Until today, I thought that my confusion was because “series” and “sequence” both begin with “s.” Now I see the real problem! Teachers would say “sum the following series.” They should have said “evaluate the following series,” since the series is already a sum.

Comment from John Chase: In non-mathematical contexts we don’t differentiate between the two. We think of “television series” and a “series” of cars in a line at an intersection. How mathematically sloppy!

Gene Chase: Yes, usually mathematical language is general language made more precise, not less precise. For example, if you tell a story elliptically, you leave things out of it; if you tell the story parabolically, you give an analog of the story; if you tell the story hyperbolically, you embellish it. The corresponding geometric figures have eccentricities which are either between 0 and 1 (ellipse), precisely equal to 1 (parabola), or greater than 1 (hyperbola).

This makes sense when you remember that “elliptic” is Greek for “defective,” “para” is Greek for “along side,” and “hyper” is Greek for “beyond.”

This is incredible! I hadn’t heard of this paradox. It reminds me of Gabriel’s horn. But I like it better, in fact, because it doesn’t require any Calculus knowledge. You know I like the harmonic series. I’ve blogged about it twice before, here and here.

I went down to the USA Science & Engineering Festival yesterday. There were thousands of people there, including our science teacher Mr. Martz at the QuarkNet booth, and our distinguished guest Glen Whitney with the Math Museum exhibit. (I also saw a few of you, too!)

A kid tries to build an unsupported arch of overlapping rectangular bricks.

At the Rockville Science Center‘s booth (the Rockville Science center doesn’t exist yet), I stopped because I immediately recognized an application of the harmonic series! The goal is to stack thin 8″ rectangular bricks in such a way that they span a gap of 22″. This girl needed a bit of my help to get started, but as you can see in the photo, now she’s doing marvelously. As I remember, the literature at the table gave instructions to overlap the top brick by half, the next by a quarter, the next by an 8th and so on. But the mathematicians in the crowd know that this overlapping strategy would limit us to a spanning distance of 16″, even given an infinite number of bricks (do you remember why?). It actually turns out that you can build this kind of stack with an infinite overlap. The overlaps are proportional to the harmonic series, which is divergent. Here’s a nice paper about it.

My origami-approximation to the hyperbolic paraboloid.

I stopped by the MAA’s booth long enough to make this origami hyperbolic paraboloid. You can learn to make your own here.

Me and Glen Whitney

Right next to the MAA’s booth was the Math Museum‘s booth. I stopped by to say hi to Glen, our speaker from the previous day. And while I was there I made a tetraflexagon (directions on their website). And I made my own Math Museum logo. Cool! Also, they have this circular laser array that allows you to see slices of solid figures. Check out my slices:

A triangular slice of the dodecahedron.

A pentagonal slice of the dodecahedron.

A slice of Mr. Chase :-).

I played around with the dodecahedron. With it you can get slices that are regular triangles & hexagons (by moving through a vertex), regular pentagons & decagons (by moving through a face), or squares & octagons (by moving through an edge). Remind anyone of Flatland? It made me curious to try some other platonic solids. My intuition is that the dual of each platonic solid would yield the same regular cross sections. But I have no idea. Anyone else know?

Mr. Chase is approximately 2 billion nanometers tall.

The last thing I did yesterday was the Nano Brothers Juggling Show. Very cool. I’ve actually seen them perform before. Those of you who know me, know I’m an avid juggler. The juggling was fun, but even more fun was the way they incorporated science into the show.

My Precalculus students have been asking me this question. I don’t really have a great answer, except that it’s true. Granted it’s not very intuitive. But nothing about infinite series is intuitive. For those not in my class or not familiar with the harmonic series, the question is:

Does converge?

And the answer is no. This is surprising, because the terms of the series approach zero.

The proof that it diverges is due to Nicole Oresme and is fairly simple. It can be found here. There are at least 20 proofs of the fact, according to this article by Kifowit and Stamps.

Interestingly, the alternating harmonic series does converge:

And so does the p-harmonic series with p>1. For instance:

Besides looking at the sequence of partial sums, I’m not sure I can help you with the intuition on any of these facts. Can you?