Why is the convergence of infinite series covered in Calculus II?

I'm teaching AP Calculus BC for the first time this year. The AP curriculum is an attempt to cover most of the material in a two-semester university freshman calculus series, and so I am reminded of my own Calculus II course.

As we finish up the curriculum, we are discussing the convergence of series. For the most part, the curriculum has a natural flow from functions to limits, then to the derivative, derivative rules and applications, integrals, integration techniques, integration and solving differential equations, improper integrals, then... infinite series representation of functions. It seems to me to be a disruption in what was otherwise a smooth progression.

I understand that for students in a math program it is very important to learn the theory of infinite series. I recall that I did not understand infinite series and convergence until taking analysis as a first semester junior.

I also observed that while the method of finding series solutions to differential equations is included in many textbooks it was not included in the undergraduate differential equations course I took, nor the few that I TA'd as a graduate student.

Why is so much attention given to infinite series in calculus II?

• One faculty in my department made a quip about switching the order and doing geometric series with Taylor series then talking about convergence is where these become defined functions. Drawing the graph of the geometric series and discussing the radius of convergence makes so much sense then. Apr 7, 2016 at 2:28
• This is the approach taken in Finney, Demana, Waits, Kennedy - Calculus: Graphing, Numerical, Algebraic ( pearsonschool.com/index.cfm?locator=PS12By ) a few of whose authors have been part of the writing committee for the AP exams. See my long-winded, independent response, too Apr 7, 2016 at 5:37
• Calc 2 does seem to be an awfully random hodge podge in many places. A bunch of integration techniques, integration "applications", then power series. Polar and parametric gets thrown in there too? Sometimes some vector stuff? It is just a place to dump a lot of stuff the engineers need, without any coherent unifying story. Apr 7, 2016 at 16:26
• Short answer: inertia. Apr 7, 2016 at 17:45
• @ChrisC Meanwhile, three weeks before the exam: "...and so that's what a convergent series is. Here's an interesting example- the function we've been relying on all year". Apr 7, 2016 at 21:41

One could make a lot of arguments either way. However, with respect to the AP curriculum as it stands, it makes sense to include infinite series, because Taylor Series are there. Why do I say this? Why include Taylor Series at this venture?

Well, the main viewpoint of the AP calculus curriculum should be local linearity. It should be how the major rules for derivates, L'Hopital's Rule, and even the FTC are justified -- not absurd limit arguments (have them see the "clever zero" somewhere else). Please see Dan Teague's resources from the North Carolina School of Mathematics or comment if you want me to post some resources for those things. Taylor polynomials naturally flow from this discussion, because you might be curious as to which functions accept a "locally quadratic" approximation. Why stop there?

Now that you've justified and played with getting ever better Taylor Polynomials and approximations for all sorts of practical purposes (including discussion of error!), one begins to conjecture what happens as the number of terms goes to infinity and bam! Convergence is necessary. Then you step back and talk about series convergence so that way you can attack all of those lovely Taylor Series with justification for their convergence to functions.

I 100000% disagree with just randomly starting with infinite series. That is not logical and is completely disconnected. It's a tool for justifying things you really want.

One could also find other reasons to talk about infinite series (not on the AP test) such as the Basel problem of Euler which gives us the lovely

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$.

I HIGHLY recommend looking at the pdf here: Basel Problem and having your class go through the document as part of the review for the AP exam. Maybe split them up into groups and have each group present on a section or something? If there isn't enough time, go over it together as a comprehensive, single problem (going through its historical development in sections 1-4 of the pdf) that requires many of the tools from the latter part of the BC curriculum. The exercises are great review problems!

Section 5 is short and after the firepower of section 4, one looks at

$\displaystyle\sum_{k=1}^{\infty}\frac{1}{k^3}$

thinking awesome! I have the tools to attack this. Wait. No I don't! Mathematics isn't dead --- no one knows how to do this, and it's been centuries since Euler solved the rather similar problem! A good way to end a BC calc course in my mind, emphasizing there is a lot of math to be learned, because it's a vibrant, living subject where a simple change in familiar, solved problems results in something unknown.

• As to other viewpoints, it is argued by many that we switch the approach and have BC calculus be what is now Calc III/Multivariable Calc, so that the entire AB/BC course contents could be unified. I wish this would be considered at all levels. I say move infinite series, improper integrals, and polar coordinates, etc to third semester calculus. My heretical (not so much) view :) Apr 7, 2016 at 5:42
• @ZachHaney Thank you for the answer. As for you comment, we have a post-AP Calc class that covers CH 11-14 of ETF first semester, then LinAlg up to vector spaces and some DiffEq in the second.. I recommend to all my students that they take calc I,II,and III at university even if they are offered college credit since the AP curriculum is shallow and we tend to focus more on the tested curriculum more than the theory ($\varepsilon \text{-} \delta$, anyone?). Apr 7, 2016 at 21:58
• And yes, I would love a link to those resources. Apr 7, 2016 at 21:58
• courses.ncssm.edu/math/Talks/PDFS/icme_8.pdf is an overview of the local linearity approach (particularly focused on justification of some otherwise vaguely considered things like L'Hopital's Rule). I highly, highly recommend their textbook (the first reference at the end of the pdf) to supplement any AP text. It's fantastic. Apr 11, 2016 at 15:46
• One can talk about Taylor polynomials and the Taylor approximation without talking about Taylor series at all. May 4, 2016 at 12:25