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Approximating Functions with a Power Series : Instructor's Guide

Main Ideas

  1. Visualizing the fit of a power series approximation to a goven function.
  2. Convergence for a power series.

Students' Task

Estimated Time: 20 minutes

Students have already calculated the coefficients for a power series expansion. Students plot several terms of the expansion against the original function in order to judge how well the approximation fits the original function.

Prerequisite Knowledge

This activity is a good followup to Calculating Coefficients for a Power Series.

This activity is designed to be a soft introduction to Mathematica or Maple. The notebook/worksheet is already prepared, but is missing some information which students will need to fill in. They will also need to learn how to step through a notebooks/spreadsheet. Students should be able to calculate coefficients for a power series expansion and they need to have the series expansion for $\sin(\theta)$ available.


Activity: Introduction

No introduction is needed - students can jump right in!

Activity: Student Conversations

  1. Students have to modify the worksheet in order to plot approximations better than 3rd order. Students who are uncomfortable with Maple (or equivalent) may have a little trouble.
  2. Students are asked to determine how many terms are needed in the approximation in order to fit the $\sin{\theta}$ function from $0$ to $\pi$. Students should be encouraged to explore higher order approximations.

Activity: Wrap-up

  1. This activity leads into a nice discussion of idealizations and making approximations. The question of “How many terms do I need to keep in my approximation?” is related to the question of “What domain do I care about?”
  2. Most students at the middle division level are familiar with small-angle approximations and the example of simple harmonic motion of a pendulum. This activity illustrates nicely how small your angle must be in order for the approximation $\sin{\theta}\approx \theta$ to make sense.
  3. You can also discuss some nice sense-making activities. One such example is being able to tell if you've got the sign wrong for a particular term - if it makes the approximation worse (the approximation diverges from the original function faster than it did with fewer terms), then you may have made a sign error.


This activity is part of a sequence of activities addressing Power Series and their application to physics. The following activities are part of this sequence.

  • Preceding activities:
    • Recall the Electrostatic Potential due to a Point Charge: This small whiteboard question has students recall the basic expression for the electrostatic potential due to a point charge which is used to begin a classroom conversation regarding what is meant by $\frac{1}{r}$.
    • The Distance Between Two Points - Star Trek: This kinesthetic activity has students work together to resolve a given problem using geometry which opens the class to a discussion about position vectors and how to generalize the $\frac{1}{r}$ factor to $\dfrac{1}{|\vec{r}-\vec{r}'|}$.
    • Calculating Coefficients for a Power Series: This small group activity has students work out the expansion coefficients of a familiar function, $\sin(\theta)$, which gives them more experience working with power series.
  • Follow-up activity:
    • Electrostatic Potential Due to Two Point Charges: This small group activity has students apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges.

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