Small Group Activity: Calculating Coefficients for a Power Series

Static Fields 2024

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

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

Students use this formula to compute the power series coefficients for a \(\sin\theta\) (around both the origin and (if time allows) \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.

What students learn
  • How to calculate the coefficients of a power series from the canonical formula \(c_n={1\over n!}\, f^{(n)}(z_0)\)
  • The meaning of basic power series vocabulary: order, coefficient, "around" a point, etc.

Power Series Coefficients

Consider the power series:

\[f(z) = \sum_{n=0}^{\infty} c_n (z-z_0)^n\]

expanded around the point \(z_0\). The coefficients are found from the formula:

\[c_n = \frac{f^{(n)}(z_0)}{n!}\]

  1. Find the first four non-zero coefficients for \(\sin\theta\) expanded around the origin.


  2. Write out the series approximation for \(\sin\theta\) expanded around the origin including only the first four non-zero terms.


    \(\sin\theta \approx \)


  3. Plot \(\sin\theta\) and your series approximation term by term and notice where the approximation is good.


  4. Find the 4th order coefficients for \(\sin\theta\) expanded around \(\theta_0 = \pi/6\).


  5. Write out the series approximation, correct to 4th order, for \(\sin\theta\) expanded around \(\theta_0 = \pi/6.\)


    \(\sin\theta \approx \)


  6. Plot \(\sin\theta\) and your series approximation term by term and notice where the approximation is good.


  7. What does it mean to write a series expansion around the point \(a\)?


  8. Briefly describe in words how to expand the series approximation for a function, correct to 4th order.


Instructor's Guide

Introduction

Start with a brief lecture giving the form of power series expansions, introducing the basic vocabulary, and deriving the formula for the coefficients. See for example, the content in Power Series. Then pass out the handout, to be completed in small groups. It is important that all students get to through the first two questions. Fast groups can go on to the later questions.

Student Conversations

  • Pay attention to the name of the independent variable. "The equation for the coefficients is given in terms of the variable \(z\). What is the independent variable in \(\sin\theta\)?"
  • 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 \(\theta=0\). If you are asked to find the power series expansion around \(z=a\), then you must plug the number \(a\) into all of the derivatives.
  • We find that once the coefficients are calculated, some students want to add the coefficients together rather than multiplying them by the appropriate monomial: \[ \cancel{f(z) = c_0+c_1+c_2+...}\] Students should be encouraged to write out the general form of the expansion in terms of the coefficients: \[ f(z) = c_0+c_1\,(z-z_0)+c_2\,(z-z_0)^2+...\]
  • Some students don't understand the notation \(f(z_0)\) as “evaluate the function at \(z=z_0\)”.
  • Some student aren't familiar with the convention that \(z_0\) is a particular value of the variable \(z\).

Wrap-up

In a whole class wrap-up, make sure to address all of the points in Student Conversations, above, especially the vocabulary. This activity is designed to be followed by the activity: Visualization of Power Series Approximations which let's students explore the graphs of the different orders of power series approximations that they have calculated in this activity.
Author Information
Corinne Manogue
Keywords
Taylor Series Coefficents Power Series
Learning Outcomes