My Favorite One-Liners: Part 35

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

Every once in a while, I’ll discuss something in class which is at least tangentially related to an unsolved problems in mathematics. For example, when discussing infinite series, I’ll ask my students to debate whether or not this series converges:

1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \dots

Of course, this one converges since it’s an infinite geometric series. Then we’ll move on to an infinite series that is not geometric:

1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \dots,

where the denominators are all perfect squares. I’ll have my students guess about whether or not this one converges. It turns out that it does, and the answer is exactly what my students should expect the answer to be, \pi^2/6.

Then I tell my students, that was a joke (usually to relieved laughter).

Next, I’ll put up the series

1 + \frac{1}{8} + \frac{1}{27} + \frac{1}{64} + \dots,

where the denominators are all perfect cubes. I’ll have my students guess about whether or not this one converges. Usually someone will see that this one has to converge since the previous one converged and the terms of this one are pairwise smaller than the previous series — an intuitive use of the Dominated Convergence Test. Then, I’ll ask, what does this converge to?

The answer is, nobody knows. It can be calculated to very high precision with modern computers, of course, but it’s unknown whether there’s a simple expression for this sum.

So, concluding the story whenever I present an unsolved problem, I’ll tell my students,

If you figure out the answer, call me, and call me collect.

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