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## Calculating Coefficients for a Power Series: Instructor's Guide

### Main Ideas

- Using a power series to approximate a function
- Understanding series notation

### Students' Task

*Estimated Time: 30 minutes*

To calculate the coefficients and terms in a power series expansion of $\sin{\theta}$. Students expand the function around two different points and to various orders of approximation.

### Prerequisite Knowledge

Students should have some familiarity with series notation and should be able to differentiate.

### Props/Equipment

- Tabletop Whiteboard with markers
- A handout for each student

### Activity: Introduction

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

$$c_n={1\over n!}\, f^{(n)}(z_0)$$

### Activity: Student Conversations

- Make sure students switch from $z$ to $\theta$!

- Emphasize the difference between the
*order*of a power series and the number of*nonzero terms*.

- Emphasize the difference between
*expanding at*$\theta=\pi/6$, and replacing $\theta$ by $\theta-\pi/6$ in the series expansion at $\theta=0$.

### Activity: Student Conversations

- At the middle division level, students initially have trouble working with equations that have a lot of symbols. In this activity, students have particular difficulty identifying the variable that they're expanding in. They don't understand the relationship between the $f(z)$ given on the worksheet and the $\sin{\theta}$ that they're given to expand.
- We find that once the coefficients are calculated, some students want to add the coefficients together rather than multiplying them by the appropriate monomial. Students should be encouraged to write out the general form of the expansion in terms of the coefficients:

$$ f(z) = c_0+c_1\,(z-z_0)+c_2\,(z-z_0)^2+…$$

### Activity: Wrap-up

In a whole class discussion, talk about some of the common difficulties you saw while walking around the room, and how to resolve those difficulties.

### Extensions

This activity is a natural lead in to Approximating Functions with a Power Series

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 $\dfrac{1}{|\vec{r}-\vec{r}'|}$.

- Follow-up activities:
- Approximating Functions with a Power Series: This computer visualization activity using Mathematica (or Maple) fits power series approximations of a given function to an actual function which allows students to see where approximations are valid.
- 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.