Activity: Calculating Coefficients for a Power Series

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:

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

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

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

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

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.


Author Information
Corinne Manogue
Learning Outcomes