More Calculus Fun with Series

Many many thanks to the wonderful Mike Thayer (@mthayer_nj) who sent a link to this lovely video in response to yesterday’s post.

We started each Calc BC class today by revisiting the rational function that caused me so many problems yesterday. Innocently enough, I decided that it would be interesting to examine 1/(1-x) first by long division where the divisor is 1 – x and we got the power series 1 + x + x^2 + x^3 + … as the quotient. Then, we used -x + 1 as the divisor and got the series -1/x – 1/x^2 – 1/x^3 – 1/x^4 … as our quotient. Class ended too soon and I was not able to answer the question of how we could consider these two very different looking series as being equal to each other since they were each results of considering the rational function 1/(1-x).

So, after sending out my call for wisdom in yesterday’s post I went to GeoGebra and discovered something lovely. Something that I was sure my students would be able to discover for themselves. I created a GGB file and I planned our day today. Can I tell you how proud I am of my students for how they handles today? Well, I won’t wait for your permission, I will just come out and tell you how thrilled I am.

I started the day by quickly revisiting yesterday’s two division results and then I called up the GGB file with only the rational function showing. They saw that I had each of the other two series expressions typed in already (out to x^5 for each) and I asked them which of the two they wanted to see first. My morning class wanted to see the series with x as the ratio first, my after lunch class wanted to see the one with the ratio of 1/x first. In each case, after unveiling the function of choice and noticing the relationship between the rational function and this new series expansion my students made the following observations:

  1. The graphs only seem to match over a certain set of x values
  2. If I were to add more terms, that match would improve
  3. If we look at the graphs of both series then we will have a nearly complete match to the graph of the rational function

What my students realized – what I realized last night – is that the two series we can generate have completely opposite intervals of convergence. It was absolutely lovely to see geogebra help this intuition along and it was fantastic that they made this realization before seeing the second graph to confirm it.

After this breakthrough we watched the video together and all of our brains were a bit achy by the end. Amazing that Mike found/knew this link so quickly when I blogged last night.

Some other notes – A student in my after lunch class made this observation about the complementary nature of the intervals of convergence before we even looked at geogebra. I gave one last example, f(x)= x^3/(x+5) and the student who was bothered by -1 = 1 + 2 + 4 + … quickly converted x + 5 into a form of 1 – r so we could interpret the rational function as an infinite geometric series. Another student converted it to x^2/(1 + 5/x) and we, once again, had two different series’ that had complementary intervals of convergence. I have taught this course five or six years before this and never had this ‘discovery’ pop up. Now, we cannot avoid it. It provides a wonderful context for our upcoming conversations about Taylor series and it gives us the opportunity to be more aware of convergence expectations. A pretty great day!

A Calculus Conundrum

Today we returned to school after a too long two week spring break and in Calculus BC we are beginning to engage with power series ideas. I decided to borrow and modify an exercise in our teacher resources binder. The text presents a definition of a power series and draws comparisons between them and geometric series. I decided to present four series examples and four answer choices – A) Geometric Series  B) Power Series   C) Both  D) Neither. My idea was to show the answers without the definition first and draw out a definition from those clues. That part of class worked pretty well. Then I dipped into turning a rational function (f(x) = 1/(1-x)) into a power series with a brief discussion of why we would even want to do such a thing. I showed them that this function is equal to 1 + x + x^2 + x^3 + … in three different ways. But before I could launch into this conversation I was challenged by one of my very talented students. He said that this equation could only be true if -1 < x < 1. I replied that this was a condition for convergence of an infinite geometric series but I did not want to wander into a conversation (yet) about radius of convergence so I tried to put off that conversation. To his credit he asked me to explain the equation if x had the value of 2 which would then mean that -1 = 1 + 2 + 4 + 8 + …  I did my best to congratulate this observation while still holding off a bit on talking about the radius of convergence.

So, I showed the comparison to the sum formula for an infinite geometric series and then I multiplied each side of the above equation by the denominator 1 – x showing that the series on the right telescopes into 1. Then I got myself in trouble. I did a long division to show that 1 / (1-x) expands into the infinite series. However, when I was writing notes to myself earlier int he morning I thought that my students would question why I was writing the divisor in the form 1 – x. Students are SO used to seeing expressions in a more standard form of -x + 1, so I also did the long division that way. Well, the result of the long division is pretty radically different. Instead of 1 + x + x^2 + x^3 + … the result is now -1/x – 1/(x^2) – 1/(x^3) -… I was pretty surprised by this and I thought that it was important to share this surprising difference. Well, I would have felt better about that decision if we had had about 5 more minutes to talk. Instead we were faced with the end of class and my students asked if these two drastically different expressions should be considered identical.

I conferred with another calculus teacher at my school and he was pretty surprised/intrigued by this conversation and pressed me a bit on why I wanted to open that door of rewriting the division. He also pointed out – as I should have – that the second response is not a power series since it has negative exponents. If I had pointed that out before class dismissed I would have felt better about our conversation. I need to do some thinking about how to talk about this tomorrow. Hey internet – any words of wisdom about this?