If a ball bounces an infinite number of times, it must take an infinite amount of time to finish bouncing!
This obvious “fact”
motivates our module on infinite series.
Background:
The ideas in this module were developed for a combined math-physics
course. Before beginning infinite
series towards the end of the second semester of calculus, we want to create
“cognitive dissonance” in our students, who otherwise think they intuitively
understand processes involving infinity. The
bouncing ball geometric series is a nice example related to Zeno’s paradoxes
that forces students to think about how infinitely many discrete steps can sum
to a finite answer. In
the combined math-physics class, we also use this module to review kinetics
and energy.
We actually use this example
in conjunction with the “tower of bricks” harmonic series example, and the
“towers of exponents” with √2 versus √3.
These three examples create the “cognitive dissonance” which probes
our students’ understandings of infinity.
The module
is a web version of our in-class hands-on activity.
Please view the module first then use the laboratory notes to implement
the hands-on lab in your own classroom.
Laboratory Notes for the Bouncing Ball Module
We also have used the many proofs of the geometric series sum in a sophomore level Foundations of Mathematics course.
Using Proofs of the Geometric Series in a Foundations of Math Course
copyright: Besson and Styer, Villanova University 12/22/2007