Student handout: Visualization of Power Series Approximations

None 2023
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.
What students learn
  • Visualizing the fit of a power series approximation to a given function;
  • Visualizing how the fit of a power series improves as more terms are added;
  • What it means to expand a power series around a point other than \(z=0\).

Use the Sage code in the activity at this link or the attached Mathematica Notebook to plot the function \(\sin\theta\) and power series approximations to the function to explore how well the approximations work.

You will first need to calculate the coefficients of the power series. For the first part of the worksheet, calculate these coefficients for the power series around \(\theta=0\) and for the second part of the worksheet, calculate the coefficients around \(\theta=\frac{\pi}{6}\). You may have done these calculations in Calculating Coefficients for a Power Series.

You will need to know a few things about Mathematica Notebooks:

  • To select a line of code, click anywhere on the line.
  • To evaluate the line of code, first select it and then hit SHIFT/ENTER.
  • Some of the lines of code are missing information (the values of the coefficients. Enter them BEFORE evaluating the line of code.)

Author Information
Corinne Manogue
Keywords
Taylor series power series approximation
Learning Outcomes