Wednesday, March 18, 2009

Analysis of a Series: An Investigation before the AP Calculus BC Exam

\displaystyle \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}

The remarkable identity above could be the subject of many math blog posts but we will look at a variation, one that is accessible to precalculus and calculus students. With the AP Calculus BC Exam looming, the following investigation can be used to introduce or to review the topic.

I'm not sure if I have ever made it really clear on this blog that I routinely used these kinds of investigations in the classroom. For those who wonder how I could possibly have completed the required coursework for the AP Calculus BC syllabus or who might question my sanity, a couple of points here:

(1) Of course I didn't do this every day. I might have done an extensive investigation once per unit.
(2) Imagine my surprise when I first saw the Finney, Demana, Waits and Kennedy text, a book that has these kinds of explorations in every chapter! I thought they had found my old lesson plans.
(3) Most of the extensive investigations were assigned for work outside the classroom. In fact, for a while, the first investigation of the year was posted on my web site and emailed to students at the end of August before they arrived in school (I met them in June before they left for the summer or I got their phone numbers from guidance and called each of them to tell them to look for the assignment online, and to download and print it.)
(4) Even if I didn't prepare an exploration every day, most every lesson plan which introduced a new topic included a series of leading questions like these. My intent was always to have them think more deeply about a topic, i.e., to understand

  • the historical origins of the topic
  • how it was connected to their prior learning
  • its usefulness and application
  • why a method or theorem works (derivation, justification)
Developing these lessons initially was labor-intensive but a work of love. Perhaps no more laborious than what another of my colleagues did for his students: developing a PowerPoint presentation for every lesson for the entire year. As time-consuming as all of that sounds, once you've done a few of these, they start to flow naturally and the following year you only have to revise!

A Series Investigation

Consider the following finite series:


(a) Write the series using summation notation.

(b) Verify the following identity for n > 1:

\frac{1}{n^2-1} = \frac{1}{2}(\frac{1}{n-1}-\frac{1}{n+1})

(c) Use the identity in (b) to show that the value of the series above is

Hint: What was Galileo's most famous invention?

(d) Using a method similar to (c) verify the following for n, even:


Note: If n = 2, the right side would be accurate however the left side would consist of only one term. I could have used summation notation for the left side but I didn't want to give away the answer to part (a).

(e) If n is odd, show that the series on the left of part (d) can be written:


(f) Show that the expression on the right side of the equation in (d) and the expression in (e) are algebraically equivalent.

(g) Use the expressions from (d) and (e) to show that the sum of the following infinite series is 3/4:


(h) There are many ways (p-series, integral test, etc.) to prove that the series \displaystyle \sum_{n=1}^\infty\frac{1}{n^2}

converges. However, for this exploration, we will use the convergence of the series in (g) to do this:
Demonstrate that this series converges using both the Comparison Test and the Limit Comparison Test by using the series in (g).

  • More commonly, the convergence of the series in (g) is demonstrated by comparing it to the p-series. We're doing the reverse here.
  • Another important aspect for precalculus and calculus students is to have them compare the partial sums to the sum of the infinite series. Thus, it's worth taking the time to have them see how close the sum is to 0.75 when adding the first 100 terms, the first 1000 terms etc. Also, indicate that the difference can be thought of as the "error" in the approximation. All of this is needed for further study and it deepens their understanding of infinite series.
  • As indicated above, this investigation may be too time-consuming for a regular period of 40-45 minutes. I would recommend doing parts (a)-(c) (or (d)) in class and assigning the rest for homework to be collected after 2-3 days.
  • Teachers of precalculus can use parts of this investigation when developing the concepts of series. Much of the groundwork for infinite series can be laid before students get to calculus!

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