We spent about six weeks on sequences and series in Calc BC. It was an interesting struggle for me personally as I last touched on the topic in 2003 taking Calculus II. The subject appears really intimidating, just search "convergence flowchart" and see what I mean. In the end, it was not nearly as complicated as I thought it would be and the kids and I got through it without any insane flowcharts.

By far the most interesting topic is Taylor/Maclaurin/power series. It's a fascinating bit of math that causes all sorts of problems, because the tiniest variation wrecks the usability of your series completely.

As we headed into Spring Break, I had the kids do a simple exploration of Maclaurin series in particular. Primarily because the series for sin/cos are so easily modified. Their objective: modify a sin/cos function in whatever way they want, iterate a Maclaurin series to four terms, name the iterations appropriately, and take a peak at their error when x = 1.

The coolest part is I don't think the kids (or myself really) was prepared for how wild their constructions could be. A few kids picked functions that worked beautifully at x = 1.

A bigger group got ok results, but had errors over 35%

And then 3 of them picked functions that just went off the rails (the joys of exponents)

450% was hardly the worst. Another was off by 22000+% and the best ever was the student whose error was infinite.

This was a nice, simple project, and I think a nice little eye-opener about how math is sometimes just a shot in the dark.