Fun with Sequences

We started the chapter on Sequences and Series in my calc II class last week. This is the chapter I have been looking forward to the most, even though, of course, all of the chapters are interesting. I decided to use one class period mostly talking about things that didn't matter for the class at all, because they are irresistibly fun.

I started with some limits of recursive sequences, which actually are (a small) part of the class. Things like

$\sqrt{2}$, $\sqrt{2\sqrt{2}}$, $\sqrt{2\sqrt{2\sqrt{2}}}$, $\ldots\rightarrow 2$

and

$a_1=1, a_{n+1}=1+1/a_n$

which converges to the golden ratio. I resisted talking too much about the golden ratio and the Fibonacci sequence, because I figured many students had probably already seen it, or could easily learn about it online (which I suggested they do). Of course, at that point, you might as well write out, without any simplifying, the first couple of terms in that sequence and start talking about continued fractions, which I did.

Then I jumped to the (unsolved) 3n+1 problem: Consider the piecewise rule that defines $a_{n+1}$ to be $a_n/2$ if $a_n$ is even, and $3a_n+1$ if $a_n$ is odd. Now given any starting number $a_1$, the question is: does the sequence starting at $a_1$ and following the piecewise rule always eventually end up at the cycle 1, 4, 2, 1, 4, 2, 1...? I really love stating these simple problems (simple to state, that is) for my class, and pointing out that they are unsolved. I've also already mentioned the problem of finding odd perfect numbers.

From there, it's probably time to draw some pictures. The Koch snowflake is a fun example to do. I only talked about how it has an infinite perimeter, and will return later to finding the area, once we've done a little more with series.

Once you're on fractals, it's hard to resist the Mandelbrot set. This one's a little harder to talk about in a calc II class, because it involves complex numbers. But you can still talk about taking a function $f$ and a starting value $a_1$, and considering the sequence $a_1, f(a_1), f(f(a_1)), f(f(f(a_1))),\ldots $. This sequence is what you consider to determine if a point is in the Mandelbrot set, using the function $f(z)=z^2+a_1$ (a different function for each point you consider). So I showed a couple printed out pictures of the set, and encouraged my students to go online and look for more pictures.

I regret, just a little, missing the Cantor set in my discussion. Perhaps another day.

That's all I got to in class. Of course, there's still several weeks left in the semester, and lots of fun things that can be said with sequences and series. When we get to Taylor series, I'll probably tell them a little about my research, which is analogous to the series they'll be looking at. Since I'll be talking about my research, I'll also talk a bit about topology.

While I was wandering around wikipedia, looking for more fun things to talk about, I ran across the page on the ? function. I'd not heard of it before, but it looks pretty interesting. I should probably look at it some more sometime.