The Fractal Tree Revisited

During a previous lesson, we used a Fractal Tree to start investigating geometric sequences and series. Today we are going to revisit this model in order to make the connection from the partial sum of a geometric series to the sum of an infinite geometric series.

This worksheet will get students thinking about the sum of an infinite geometric series and whether or not they will increase with or without a bound. I give the worksheet to my students and have them work with their table groups on it for about 10 minutes.

Question #3 is really interesting and I will usually hear some good conversations about it. At this point I will pull the class back together and we will have a discussion about how long it will take for the fractal tree to reach a height of 5 units. Here are some thoughts that came up when I taught it:

1. It will never reach a height of 5 units. The formula I came up for on the original day of the lesson for the height of the tree at stage n is 4 - 1/2^(n-2), so whatever n is, we will be subtracting a small amount from 4 to get the height. The height will never even reach 4 units, let alone 5.
2. The height has to reach 5 units. We are adding a new unit to the height every time and the height is constantly increasing, so eventually it will have to reach 5 units.
3. If we were to graph y = 4 – 1/2^(n-2), then 4 would be an asymptote. The graph would get really close to y = 4 but would never intersect the line.
4. We are adding height to the tree with every stage, but the new height being added is getting really, really small. So eventually the new heights that are added will not affect the total height anymore.

It was really interesting to see those who originally thought that the height had to equal 5 units change their mind once they were presented with some really good arguments from those who disagreed. Their conversation was a really good example of MP3 in practice!

Once students have an intuitive sense of why the height of the fractal tree will never surpass 4, I ask them what will happen if we actually did add every single term of the infinite geometric series together. In the video below, I discuss how we can use the formula for the nth partial sum of a geometric series and modify it to find out the height of the fractal tree if we added an infinite number of terms together.

After we get this new formula, I present students with four sequences (shown here and here). I tell them to use this new formula for sequence B (where r = 2) and see what happens. They will get -1/2 as the infinite sum but clearly that cannot be correct since the terms are getting larger and larger and the sum will go to infinity.

Then I ask for some conjectures about when the sum will approach a certain number and when the sum will increase or decrease without bound. I introduce the vocabulary words convergent and divergent and explain their meaning to students. Next I give students a couple minutes to decide if the sum of the four geometric sequences will converge or diverge.

Usually my students will realize when 0 < r < 1 then the sum will definitely converge to a certain value. They are unsure about negative r values, so I encourage them to actually find partial sums until they figure it out. Once students figure this out, we will amend our formula for S = a/(1 – r) to note that it only works when -1 < r < 1.