Activity: Calculating Coefficients for a Power Series

Theoretical Mechanics (8 years)

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.

• This activity is used in the following sequences
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, correct to 4th order, for $\sin\theta$ expanded around the origin.

$\sin\theta =$

3. Find the first four non-zero coefficients for $\sin\theta$ expanded around $\theta_0 = \pi/6$
4. Write out the series approximation, correct to 4th order, for $\sin\theta$ expanded around $\theta_0 = \pi/6.$

$\sin\theta =$

5. What does it mean to write a series expansion around the point $a$?
6. 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. Students should be encouraged to write out the general form of the expansion in terms of the coefficients: $f(z) = c_0+c_1\,(z-z_0)+c_2\,(z-z_0)^2+...$

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.

• computer Visualization of Power Series Approximations

computer Computer Simulation

30 min.

Visualization of Power Series Approximations
Theoretical Mechanics (13 years)

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot $\sin\theta$ simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.
• assignment Series Convergence

assignment Homework

Series Convergence

Power Series Sequence (E&M)

Static Fields 2023 (6 years)

Recall that, if you take an infinite number of terms, the power series for $\sin z$ and the function itself $f(z)=\sin z$ are equivalent representations of the same thing for all real numbers $z$, (in fact, for all complex numbers $z$). This is what it means for the power series to “converge” for all $z$. Not all power series converge for all values of the argument of the function. More commonly, a power series is only a valid, equivalent representation of a function for some more restricted values of $z$, EVEN IF YOUR KEEP AN INFINITE NUMBER OF TERMS. The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

Find the power series for the function $f(z)=\frac{1}{1+z^2}$. Then, using the Geogebra applet from class as a model, or some other computer algebra system like Mathematica or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. You may also need to play with the values of $z$ that you plot. Keep adding terms until you see a really strong effect!

Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

• assignment Power Series Coefficients A

assignment Homework

Power Series Coefficients A
Static Fields 2023 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the series expansion for $f(z)=e^{-kz}$ to second order around $z=3$.
• assignment Power Series Coefficients B

assignment Homework

Power Series Coefficients B
Static Fields 2023 (6 years) Use the formula for a Taylor series: $f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n$ to find the series expansion for $f(z)=\cos(kz)$ to second order around $z=2$.
• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2023 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
• assignment Fourier Series for the Ground State of a Particle-in-a-Box.

assignment Homework

Fourier Series for the Ground State of a Particle-in-a-Box.
Oscillations and Waves 2023 (2 years) Treat the ground state of a quantum particle-in-a-box as a periodic function.
• Set up the integrals for the Fourier series for this state.

• Which terms will have the largest coefficients? Explain briefly.

• Are there any coefficients that you know will be zero? Explain briefly.

• Using the technology of your choice or by hand, calculate the four largest coefficients. With screen shots or otherwise, show your work.

• Using the technology of your choice, plot the ground state and your approximation on the same axes.
• assignment Memorize Power Series

assignment Homework

Memorize Power Series

Power Series Sequence (E&M)

Static Fields 2023 (4 years)

Look up and memorize the power series to fourth order for $e^z$, $\sin z$, $\cos z$, $(1+z)^p$ and $\ln(1+z)$. For what values of $z$ do these series converge?

• format_list_numbered Power Series Sequence (E&M)

format_list_numbered Sequence

Power Series Sequence (E&M)

The first three activities provide an active-engagement version of the canonical mathematical and geometric fundamentals for power series. The subsequent activities apply these ideas to physical situations that are appropriate for an upper-division electromagnetism course, using concepts, terminology, and techniques that are common among physicists, but not often taught in mathematics courses. In particular students use the memorized formula for the binomial expansion to evaluate various electrostatic and magnetostatic field in regions of high symmetry. By factoring out a physical quantity which is large compared to another physical quantity, they manipulate the formulas for these fields into a form where memorized formulas apply. The results for the different regions of high symmetry are compared and contrasted. A few homework problems that emphasize the meaning of series notation are included.

Note: The first two activities are also included in Power Series Sequence (Mechanics) and can be skipped in E&M if already taught in Mechanics.

• keyboard Electric field for a waffle cone of charge

keyboard Computational Activity

120 min.

Electric field for a waffle cone of charge
Computational Physics Lab II 2022

Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2023 (8 years)

Power Series Sequence (E&M)

Warm-Up

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

Author Information
Corinne Manogue
Learning Outcomes