I saw a neat video from Gary Rubenstein recently:

In the video he presents a neat Theorem about partitions due to Euler.

Simon Gregg, by coincidence, was looking at partitions recently, too, and has written up a nice post which includes some ideas from Rubenstein’s video:

The part that struck me in Rubenstein’s video wasn’t about partitions, though, it was about the manipulation of the infinite product. It all works out just fine, which is pretty neat, but sometimes manipulating infinite quantities produces strange results. See this famous video from Numberphile, for example:

Just as an aside, here’s a longer and more detailed explanation of the same result:

The fascinating thing to me is that Euler’s proof in Rubenstein’s video is easy to believe, but the sum in the Numberphile video is not easy to believe at all. Both are examples, I think, of what Jordan Ellenberg called “algebraic intimidation” in his book How not to be Wrong. I used Ellenberg’s idea when I talked about the -1/12 sum with my kids:

Jordan Ellenberg’s “Algebraic Intimidation”

The talk I’d like to give to calculus students would start with the theorem presented in Rubenstein’s video. Once the students were comfortable with the ideas about the infinite products and the ideas about partitions, I’d move on to the idea in the Numberphile video. It would be a fun way to show students that infinite sums and products can be strange and you can sometimes stumble on really strange results.