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?
Not sure if this helps, but 1/(1 – x) can be rewritten as (-1/x)(1/(1 – (1/x))).
So if you take the series for 1/(1 – x) = 1 + x + x^2 + x^3 + … and replace x by 1/x you get
1 + 1/x + 1/x^2 + 1/x^3 + … (The interval of convergence would be |x| > 1 now, I think, but I’m tired and could be wrong)
Then multiply by (-1/x) and you will get the series you got from long division of 1 by (-x + 1).
I’m not surprised you got a different answer – when dividing 1 by 1 – x, you are trying to “cancel” the 1 with a 1, leaving you with an extra x; canceling that in the next step gets you an extra x^2, etc. When you divide 1 by (-x + 1), you are trying to cancel the 1 with an x, getting you into negative-exponent-land. Good times abound!
Meanwhile, for your student who asked if -1 = 1 + 2 + 4 + 8 + …: https://www.youtube.com/watch?v=XFDM1ip5HdU
Thank you for the quick and smart reply Mike. I am beginning to feel better about having a meaningful follow up conversation tomorrow. I did a Geogebra sketch and see that the intervals of convergence are (unsurprisingly) opposites of each other.