From our work with geometric series, we knew that if |x| < 1
A complete side note...I'm using Text the World to embed all the mathy stuff (thanks Sue). While I've found it to be a bit buggy and clunky, it supposedly will work in blogger, google docs, gmail, etc. so I'm excited about the possibility. This is my first attempt to use this so let me know if you're having trouble reading anything.
Anyway, from an algebraic perspective, I think this is quite surprising. Basically, we're saying that this infinite polynomial is equal to a nice "simple" rational expression.
So is this true for other functions?
I asked my students what functions would be especially interesting to look at.
is not a terribly interesting expression to write as a polynomial. sin(x) on the other hand...that would be curious.
Here's what they came up with as a class:
- trig functions
- log functions
Me: So is a particularly interesting function in calculus because...
Class: it's derivative is also
Me: Great. Try and find a polynomial with this same property.
They broke into small groups and began exploring. Every group quickly realized that a finite polynomial won't cut it. They then began trying different infinite polynomials. I gave a few groups some guidance around starting with and then trying to find the coefficients that would make the property of f(x)=f'(x) hold. Different groups approached this in different ways. Some differentiated to get and then set the coefficients of each power to be equal to each other. One group did something similar with the integral instead of the derivative. Another group did something similar to this, but in more of an informal way where they realized that as you repeatedly differentiate you'll end up with factorials in from of your coefficients and then played around with some different polynomials with factorials in them.
I didn't help at all. To my students' credit, when I asked 15 minutes later if they wanted us to come back together and talk/get hints/share ideas. I got a resounding no. Woo hoo!
They ended up working independently for about 30 minutes. Every group found a polynomial that worked. Some groups realized that any multiple of their solution would have the property that f(x)=f'(x). Two groups went further and attempted to define some equality between and their polynomial. One individual began working to find a polynomial with derivative characteristics to sin(x) ie f(x)=-f''(x). Anyway, everyone independently came up with some version of
I was pretty excited about this.
Sure, we still have work to do in terms of formalization to define a more robust definition of equality (more than just having two functions with the same property). It'll be interesting, though, to see if this helps students understand MacLaurin Series and Taylor Series.