Alternate ending: draft of a new Chapter 8

I’ve written a rough draft of a very different version of Chapter 8 for Active Calculus (single variable), and I am interested in feedback from users.

TL;DR version: I have come to think that the Taylor series representations of the sine, cosine, and exponential functions are the most important series-related ideas in calculus 2, and the details of convergence tests are not. So I wrote a chapter with its central focus on the former, and very limited discussion of the latter.

Taylor polynomials provide an intuitive way to see why infinite series are important, and their approximations of non-polynomial functions are a powerful application of calculus. Rather than me write on at length about my motivations and the changes, I think the draft speaks for itself, so I encourage anyone interested to check it out: alternate Chapter 8.

While the entire book is found at the above link based at my home page, please know that the definitive current version is always hosted at https://activecalculus.org/single.

Three additional related things:

+ I am teaching calculus 2 this fall and will be using this new Chapter 8 with my own students. In the very near future I will post an invitation to anyone who wants to do a “teach-along” with me this semester.
+ I realize the Chapter 8 draft is rough and needs work: prose editing, mathematical details, more and better exercises, and likely some more images. Feedback is welcome on any of it. I plan to make it much tighter by November 1, which is a couple of weeks before I will share it with my students.
+ I plan to (eventually, not this semester) write one more section and probably also an appendix. The additional section will likely be on using series to solve differential equations, and the appendix would be on some of the details of series convergence that are intentionally omitted in the new Chapter 8.

If you have comments or suggestions on the draft of Chapter 8, please email them to me directly: boelkinm at gvsu dot edu. If you think you will plan to use it with your students, I would welcome knowing that as well, as then I can communicate directly about updates and more with the people who most need to know.

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