The Maclaurin series for the function $f$ is given by $$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } = 2 x + \frac { 4 x ^ { 2 } } { 2 } + \frac { 8 x ^ { 3 } } { 3 } + \frac { 16 x ^ { 4 } } { 4 } + \cdots + \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } + \cdots$$ on its interval of convergence. (a) Find the interval of convergence of the Maclaurin series for $f$. Justify your answer. (b) Find the first four terms and the general term for the Maclaurin series for $f ^ { \prime } ( x )$. (c) Use the Maclaurin series you found in part (b) to find the value of $f ^ { \prime } \left( - \frac { 1 } { 3 } \right)$.
The Maclaurin series for the function $f$ is given by
$$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } = 2 x + \frac { 4 x ^ { 2 } } { 2 } + \frac { 8 x ^ { 3 } } { 3 } + \frac { 16 x ^ { 4 } } { 4 } + \cdots + \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } + \cdots$$
on its interval of convergence.
(a) Find the interval of convergence of the Maclaurin series for $f$. Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for $f ^ { \prime } ( x )$.
(c) Use the Maclaurin series you found in part (b) to find the value of $f ^ { \prime } \left( - \frac { 1 } { 3 } \right)$.