The function $f$ has a Taylor series about $x = 2$ that converges to $f(x)$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 2$ is given by $f^{(n)}(2) = \frac{(n+1)!}{3^n}$ for $n \geq 1$, and $f(2) = 1$. (a) Write the first four terms and the general term of the Taylor series for $f$ about $x = 2$. (b) Find the radius of convergence for the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer. (c) Let $g$ be a function satisfying $g(2) = 3$ and $g'(x) = f(x)$ for all $x$. Write the first four terms and the general term of the Taylor series for $g$ about $x = 2$. (d) Does the Taylor series for $g$ as defined in part (c) converge at $x = -2$? Give a reason for your answer.
The function $f$ has a Taylor series about $x = 2$ that converges to $f(x)$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 2$ is given by $f^{(n)}(2) = \frac{(n+1)!}{3^n}$ for $n \geq 1$, and $f(2) = 1$.
(a) Write the first four terms and the general term of the Taylor series for $f$ about $x = 2$.
(b) Find the radius of convergence for the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.
(c) Let $g$ be a function satisfying $g(2) = 3$ and $g'(x) = f(x)$ for all $x$. Write the first four terms and the general term of the Taylor series for $g$ about $x = 2$.
(d) Does the Taylor series for $g$ as defined in part (c) converge at $x = -2$? Give a reason for your answer.