The Maclaurin series for a function $f$ is given by $$\sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } = x - \frac { 3 } { 2 } x ^ { 2 } + 3 x ^ { 3 } - \cdots + \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } + \cdots$$ and converges to $f ( x )$ for $| x | < R$, where $R$ is the radius of convergence of the Maclaurin series. (a) Use the ratio test to find $R$. (b) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime }$, the derivative of $f$. Express $f ^ { \prime }$ as a rational function for $| x | < R$. (c) Write the first four nonzero terms of the Maclaurin series for $e ^ { x }$. Use the Maclaurin series for $e ^ { x }$ to write the third-degree Taylor polynomial for $g ( x ) = e ^ { x } f ( x )$ about $x = 0$.
The Maclaurin series for a function $f$ is given by
$$\sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } = x - \frac { 3 } { 2 } x ^ { 2 } + 3 x ^ { 3 } - \cdots + \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } + \cdots$$
and converges to $f ( x )$ for $| x | < R$, where $R$ is the radius of convergence of the Maclaurin series.
(a) Use the ratio test to find $R$.
(b) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime }$, the derivative of $f$. Express $f ^ { \prime }$ as a rational function for $| x | < R$.
(c) Write the first four nonzero terms of the Maclaurin series for $e ^ { x }$. Use the Maclaurin series for $e ^ { x }$ to write the third-degree Taylor polynomial for $g ( x ) = e ^ { x } f ( x )$ about $x = 0$.