Error: I Approximate-Actual I
11) Let \( f \) be a function with derivatives of all orders and for which \( f(2)=7 \). When \( n \) is odd, the \( n \)th derivative of \( f \) at \( x=2 \) is 0 . When \( n \) is even and \( n \geq 2 \), the \( n \)th derivative of \( f \) at \( x=2 \) is given by \( f^{(n)}(2)=\frac{(n-1)!}{3^{n}} \).
a) Write the sixth-degree Taylor polynomial for \( f \) about \( x=2 \).
\[
f(6)(2)=\frac{5!}{36}
\]
\[
\begin{array}{l}
f(2)(2)=\frac{1}{6}+f(4)(2)=\frac{3!}{3^{4}} \rightarrow \frac{2}{3^{3}} \rightarrow \frac{2}{27} \\
\frac{7+\frac{1}{6}(x-2)^{2}}{2}+\frac{\frac{2}{87}(x-2)^{4}}{4!!}+\frac{\frac{5!}{36}(x-2)^{6}}{6!} \\
\rightarrow 7+\frac{1}{12}(x-2)^{2}
\end{array}
\]
b) In the Taylor series for \( f \) about \( x=2 \), what is the coefficient of \( (x-2)^{2 n} \) for \( n \geq 1 \) ?
c) Find the interval of convergence of the Taylor series for \( f \) about \( x=2 \). Show the work that leads to your answer.