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