Now that we've talked about geometric series, it's possible to compute the area of the Koch snowflake, whose length we already computed in class when we talked about sequences. Our text (Stewart) has an exercise about the Cantor set, and the Sierpinski Carpet. The students are asked to compute the total length removed from [0,1] in order to make the Cantor set, and similarly the area removed to make Sierpinski's Carpet. I hope they found it surprising that the bits removed had the same length as the interval, and area as the square, even though there are still (clearly) infinitely many points remaining. While I was at it, I mentioned the Menger Sponge origami projects (and looking them up again convinced me that I should probably do one someday).
Though I've not yet figured out how, exactly, to bring it up in class, I think we're going to have a day and talk about cardinalities a little. I should at least show them that there is more than one infinity. Clearly it ties in to the examples from the previous paragraph, but it's also just good for their general education. Plus it's hugely fascinating. And they've spent all this time thinking about functions on the real line, they might as well know a little more about the line itself.
Even though I don't know a whole lot about it, since we were talking about p-series the other day I figured I should mention Riemann's ζ function, the associated Hypothesis, and the million dollar reward for a proof. They seemed to enjoy that there was a financial incentive, but at least one commented that there were surely easier ways to make a million dollars. I also showed how the divergence of the harmonic series implies that there are infinitely many primes, by considering the expression
$\sum_{n=1}^{\infty}\frac{1}{n}=\prod_{p}\frac{1}{1-p^{-1}}=\prod_{p}\left(\sum_{k=0}^{\infty} 1/p^k\right)$
These products are taken over all primes.
We also talked about the integral test in class, so I couldn't help bringing up the Euler-Mascheroni constant γ. I didn't say horribly much about it in class, but did mention how it can be used to approximate the number of terms in the harmonic series you would have to add up in order to get a chosen value. I worked through determining that it requires more than 12000 terms to just add up to 100.
The next thing I plan on talking about in class is sometimes referred to as the Kempner Series, and is obtained from the harmonic series by removing those terms whose denominators have some chosen digit. For example, remove all the terms that have a 0 as a digit in the denominator. What is initially surprising about this series is that it converges, though after some more thought you realize you've thrown out rather a lot of the series. Anyway, to show the Kempner series converges, you compare against a geometric series - all things we're talking about in my class.
Perhaps some of the most fun (besides, perhaps, the multiple infinities) will be once we get to Taylor series. I plan on presenting one of Euler's proofs that $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, by thinking about the Taylor series for $\sin(x)/x$. The way this works is to realize that $\sin(x)/x$ is zero at the non-zero integer multiples of $\pi$, so you think of the infinite polynomial that is the Taylor series as also being the product $\prod_{0\neq n}(1-x/(n\pi))$. After expanding this infinite product, you compare the coefficient of $x^2$ to the corresponding coefficient from the Taylor series, and Robert's your father's brother.
Another fun example I plan on doing (at least to some extent) is showing that
$\int_0^1 \frac{1}{x^x}dx=\sum_{n=1}^{\infty}\frac{1}{x^x}.$
This requires a bit of messing about, but it's so pretty that it's got to be worth it, right?The final fun thing about Taylor series is it gives a nice segway into my research. Of course, I won't get too involved with telling them about my research, exactly. But I will mention that it's a direct analogy to Taylor series. And it'll give me a chance to talk about topology, which is always good.
Update 20081103: I realized I should include some text references here, sorry about that. The two books I had in mind were:
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