I’ve been a little busy both at home and at work this week and as of this morning hadn’t given any thought to our Family Math projects for this weekend. More or less on a whim I decided to return to an old favorite topic for today’s project – infinite series, and specifically geometric series.
Two of our prior talks on infinite series are linked below, you can find others on the blog under the tag “infinite series”:
The first part of the talk today was introducing the concept of a geometric series. The main idea I’m trying to get at today is showing how we can extend a common way of showing why 0.9999… = 1 to the problem of summing a geometric series. We talk through some of the basic ideas using the series 1 + 1/2 + 1/4 + 1/8 + . . . as our example.
The next thing that we looked at was the series 1 + 1/3 + 1/9 + 1/27 + . . . . My older son initially believes that this series will also sum to 2 because it goes on forever. My younger son’s initial guess is that it will sum to 1.5. His reason is that (except for the first term) the terms are smaller than the series 1 + 1/2 + 1/4 + 1/8 + . . . .
One theme that shows up here that will continue for the rest of today’s project is that subtracting two infinite series is a little confusing to the boys. I should have found a better way, or at least an alternate way, to explain this idea to them.
In the next talk I wanted to have the boys pick their own series to sum. Unfortunately, I wasn’t clear with them that I wanted to look only at series where the terms went to zero. That lack of clarity caused a small problem at the start of this part of the project.
Once we got on the right path, we worked through the series 1 + 1/5 + 1/25 + 1/125 + . . . without too much difficulty. But the next series caused a little bit of trouble:
1 – 1/3 + 1/9 – 1/27 + . . . .
The subtraction and the negative signs were big stumbling blocks here. I really needed to provide a better way to help them see what was going on when we subtract one series from another.
In the next part of our talk we moved on to talking about an general geometric series. This discussion is a big step up in abstraction. I think this abstraction was not as difficult for my older son as it was for my younger son, which isn’t a huge surprise. Subtracting the individual terms in each series still presented a little bit of difficulty. We did manage to get to find a fairly simple formula for our sum, though. Even with the difficulty we had, I think the discussion here are a nice example of how you can take an idea from a specific setting and use it in a slightly more abstract setting.
The last part of today’s project involved using the formula we found in the last video in the situations that we’d already considered. A few examples showed that our formula seemed to match the prior results. Yay!
We then wrapped up by looking at a few situation where the terms in the series do not go to 0. Here the formula produces some results that seem strange. For now I’m leaving these odd results as fun little paradoxes for the boys to ponder.
Watching these talks as I put this blog together makes me wish I’d done a better job with this project. I think that the important mathematical ideas here can be made accessible to kids if you present them in the right way. The results are neat and some seem strange (you’ll hear my son reference Numberphile’s video about the result 1 + 2 + 3 + . . . = -1/12 in the last video – so these strange results can really make kids think).
Hopefully the next time we return to this topic I’ll remember the lessons from this one and present some ideas in a slightly different (and hopefully slightly better!) way.