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 $x = 2$. (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.
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 $x = 2$.\\
(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.