Activity 8.6.4.
Let \(f\) be the function given by the power series expansion
\begin{equation*}
f(x) = \sum_{k=0}^{\infty} (-1)^k \frac{x^{2k}}{(2k)!}\text{.}
\end{equation*}
(a)
Assume that we can differentiate a power series term by term, just like we can differentiate a (finite) polynomial. Use the fact that
\begin{equation*}
f(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots + (-1)^k \frac{x^{2k}}{(2k)!} + \cdots
\end{equation*}
to find a power series expansion for \(f'(x)\text{.}\)
(b)
Observe that \(f(x)\) and \(f'(x)\) have familiar Taylor series. What familiar functions are these? What known relationship does our work demonstrate?
