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.
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:
\[c_n = \frac{f^{(n)}(z_0)}{n!}\]
Find the first four non-zero coefficients for \(\sin\theta\) expanded around the origin.
Write out the series approximation for \(\sin\theta\) expanded around the origin including only the first four non-zero terms.
\(\sin\theta \approx \)
Plot \(\sin\theta\) and your series approximation term by term and notice where the approximation is good.
Find the 4th order coefficients for \(\sin\theta\) expanded around \(\theta_0 = \pi/6\).
Write out the series approximation, correct to 4th order, for \(\sin\theta\) expanded around \(\theta_0 = \pi/6.\)
\(\sin\theta \approx \)
Plot \(\sin\theta\) and your series approximation term by term and notice where the approximation is good.
What does it mean to write a series expansion around the point \(a\)?
Briefly describe in words how to expand the series approximation for a function, correct to 4th order.