When we did our “math biographies” project a few weeks ago I asked the kids to tell me about something in math that they heard was true but that they do not believe is true. My younger son mentioned the seemingly strange property of the Koch Snowflake having an infinite perimeter but a finite area, and my older son brought up this series:
1 + 2 + 3 + . . . . = -1/12
The “math biographies” project is here:
Our two projects about the Koch Snowflake are here:
And a little bit of my thoughts leading up to the project today are here (including the 4 screen shots of passages in G. H. Hardy’s book “Divergent Series”):
I think that explaining to kids how the series 1 + 2 + 3 + . . . can possibly equal -1/12 is practically impossible. You might actually be able to explain the idea to high school calculus students – and it would be fun to try – but, anyway, today’s project doesn’t quite go all the way to an explanation.
Instead what I tried to do was show some other strange results with infinite series that would seem to not have a sum and then walked the boys through some of the philosophical ideas in Hardy’s book.
We started with two infinite series that they were already familiar with:
(1) 1 + 1/2 + 1/4 + 1/8 + . . . = 2, and
(2) 0.99999999….. = 1.
Listening to what both kids have to say about these two series, I’m actually pretty happy with how they understand them.
For the second part of the project I extended the idea in the “proof” that the boys used to show that 0.99999…. = 1 and applied it to the series
9 + 90 + 900 + 9000 + . . . .
They were surprised to see that using exactly the same idea that they had just used, this series “sums” to -1.
I loved that my younger son immediately recognized that this strange number:
….99999.999999…. has to equal zero!
Now me moved on to looking at geometric series and finding a close formula for the series:
Looking at the formula we found, we see (possibly surprising) explanation for the results that we’ve seen previously in this project.
I love the questions that the kids had during this part of the project.
Finally we looked at some of the passages in G. H. Hardy’s “Divergent Series” book. I wanted to go through these passages so that (i) the boys could see how a professional mathematician like Hardy thinks about these seemingly strange results, and (ii) more simply, so that the boys could see these strange results in writing.
Some of the ideas are probably a little too philosophical for the boys to understand, but I thought it would still be important for them to see them.
So, that’s my best effort 🙂 Maybe several years from now when my older son is learning Calculus, we’ll revisit the topic. It was fun to think about how to share these ideas with kids, and especially nice to hear my younger son say at the end of the last video that this project was “pretty fun.”