Just for fun – some infinite sums

I was listening to Jordan Ellenberg’s book “How not to be Wrong” on the way back from Cape Cod yesterday. This time through his short discussion on infinite series caught my attention. Since in heading to DC for an ultimate frisbee tournament this weekend, I thought I’d do our weekend Family Math a day early and talk a little bit about infinite series.

We’ve talked a little bit about infinite series before – motivated mainly by Vi Hart’s videos about why .9999…. = 1 and the Numberphile video about the sum 1 + 2 + 3 + 4 + . . . – and although this talk goes a tiny bit deeper the goal isn’t rigor, just fun. I’ll link the Vi Hart and Numberphile video at the end of this post.

I started off asking the boys about infinite series and they mentioned the two examples that they’d seen before. Neither of them seems to believe the Numberphile video which was nice to hear – at least they are thinking about why the result in the video seems strange. Next we talked about why 0.999… = 1 and a few of the common “proofs” including the one that Ellenberg refers to by the catchy phrase “algebraic intimidation.”

In the next video we sort of explore Ellenberg’s “algebraic intimidation” phrase by looking at another example from his book – the series 1 + 2 + 4 + 8 + 16 + . . . . Here we apply one of the techniques that we used in the last video to show that this series seems to have a value equal to -1. Wait – what??

We finished up with another series where the algebraic techniques we used to show 0.9999… = 1 produce a strange answer. The series that we consider here is 1 – 1 + 1 – 1 + 1 – 1 + 1 . . . . The boys arrive at the conclusion that the sum seems to be either 0 or 1. We then go through the algebra to show that you get the surprising answer of 1/2, but they are not convinced.

This was a fun little discussion. Obviously the details of infinite series are a little bit over their heads right now, but it is neat to see them thinking about results that make sense and results that don’t seem to make sense at all. One of the other neat ideas that I’ve taken away from Ellenberg’s book is understanding what ideas are “obvious” and what ideas are not “obvious.” Once mathematicians started asking questions about infinite sums, it took a couple of centuries to get their heads around the issues. It is a nice that Ellenberg is able to provide lots of examples of “obvious” results that are not obvious at all.

Finally, here are the Vi Hart and Numberphile videos just for completeness: