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In my analysis class I use the following problem to illustrate the divergence of the harmonic series (consider this as a hint for solving it).

Exercise. A beetle creeps along a 1-meter infinitely elastic tape with constant velocity. Every hour the tape is lengthened out by 1 meter, and the beetle remains at the same rate of the tape it has already reached to the moment. Will the beetle ever reach the end of the tape?

This is not a paradox but a calculation of a mathematically idealised model "from life". Do you have some other, probably nicer examples which illustrate some standard but deep results in analysis, algebra, probability, geometry, and so on, and so on, and so on. Please keep in mind an average undergraduate student as the audience for your example and allow others to use it in his/her teaching.

Thank you!

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Another popular story about the divergence of the harmonic series is, how many unit bricks can you place on top of one another without glueing them (in an idealized constant gravity field) –  Pietro Majer May 28 '10 at 8:26
Yes, this story is sometimes attributed to Karl Friedrich Hieronymus Freiherr von Münchhausen, who claimed that he has used these brick stairs to get into an enemy's fortress. –  Wadim Zudilin May 28 '10 at 8:58
Just an off-topic comment. Unless you define things very carefully, the brick example is actually incorrect. See (for example) maa.org/mathtourist/mathtourist_01_28_09.html –  Peter Shor May 28 '10 at 11:03
Your site provides terific overhang constructions but does not say that the above brick example is incorrect. It is correct for an idealised model. –  Wadim Zudilin May 28 '10 at 11:52
Raymond Smullyan wrote many books that demonstrate non-trivial results from logic through entertaining stories. –  Dan Piponi May 29 '10 at 0:45

3 Answers 3

There are all these examples surrounding fixed point theorems. The following is somehow a cliche. Take a sheet of paper, crunch it, and put it on top of its original position. Then there is a point that lies on the vertical of its previous position. This illustrates the fixed point theorem for contractions in Banach spaces.

There are also a lot of examples in probability theory. Here is one related to the harmonic series. In your youth, you may have collected cards depicting soccer players, martians, whatever. There is a finite number of cards to collect, say n. Each packet of corn flakes comes with one of them, at random. And of course you want your mom to buy this precise brand of flakes so as to get the whole collection. May be you have wondered what is the average number of packets she should buy so that you can complete the collection. The answer is

$$ n \sum\limits_{k=1}^n {1 \over k} $$

This is asymptotic to $n\log n + n\gamma+ 1/2$, where $\gamma$ is the Euler constant. This is the simplest mundane example I know involving that constant. So for example, if there are $n=150$ cards to collect, you need to buy an average of 519 cards.

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You have answered a question I actually never thought of, but I wish I did! :-) –  Andrea Ferretti May 28 '10 at 10:24
This assumes the manufacturer prints the same number of each card. I don't think they do this (but I don't know that). –  Bruce Westbury May 28 '10 at 11:20

How do you hold a pizza (with your hands) so that the toppings don't slide off? That is, how do you keep the pizza "straight"? You bend it through the middle and use one of your fingers to support the bend. Why does it work? Because a pizza is inherently "flat." You would expect any reasonable definition of curvature to give a (flat) pizza slice a 0. It turns out this expression (Gaussian curvature) is the product of two numbers, $\kappa_1$ and $\kappa_2$, corresponding to the two orthogonal curvatures. Thus, one of the two curvatures must be 0 -- so if you're bending one direction, the other must necessarily be "straight" (no curvature).

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I once thought of the following variant of your beetle problem. Suppose a kid has a piggy bank with a dollar in it and gets an allowance of a dollar a week (so, this story is implausible). Then, of course, his savings grow without bound. If he has $n$ dollars, then the next week he receives a fraction $1/n$ of his savings, and so $1 + 1/2 + 1/3 + \dots$ must also grow without bound. Tell this to your calculus students but don't analyze it -- it requires infinite products!

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All this shows is that 1 + (1/2)(2) + (1/3)(3) + ... grows without bound. –  Gabe Cunningham Jul 25 '10 at 13:58
Depends how you look at it (anyway, it's intended as an intuitive argument). It says that $(1 + 1)(1 + 1/2)(1 + 1/3)\dots$ is infinite. An infinite product of this form converges if and only if the associated series $1 + 1/2 + 1/3 + \dots$ converges absolutely. –  Ryan Reich Jul 25 '10 at 14:15
Yes, but the direction you're using this in is the non-obvious one. I think you run a serious risk of confusing the best students in the class (those who would bother to check what the argument you're making really looks like) with this comment. –  JBL Jul 25 '10 at 14:58
I agree that it's not obvious, and maybe not a good idea to use in class. I still think it works okay intuitively, as long as you keep the mathematician hat off, because of how people think of cumulative percentages (and if you put the hat back on, the notion that cumulative percentages are additive is exactly this theorem). –  Ryan Reich Jul 25 '10 at 19:32
Thanks, Ryan! To be honest I didn't plan the question to be around the harmonic series (and its divergence) only. It was aimed at seeing some other examples from real analysis which, in a certain sense, contradict a straightforward intuition. You answer definitely falls into this category. :-) –  Wadim Zudilin Jul 25 '10 at 23:46

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