The Maclaurin series for $\ln ( 1 + x )$ is given by $$x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } + \cdots + ( - 1 ) ^ { n + 1 } \frac { x ^ { n } } { n } + \cdots$$ On its interval of convergence, this series converges to $\ln ( 1 + x )$. Let $f$ be the function defined by $$f ( x ) = x \ln \left( 1 + \frac { x } { 3 } \right)$$ (a) Write the first four nonzero terms and the general term of the Maclaurin series for $f$. (b) Determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer. (c) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Use the alternating series error bound to find an upper bound for $\left| P _ { 4 } ( 2 ) - f ( 2 ) \right|$.
The Maclaurin series for $\ln ( 1 + x )$ is given by
$$x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \frac { x ^ { 4 } } { 4 } + \cdots + ( - 1 ) ^ { n + 1 } \frac { x ^ { n } } { n } + \cdots$$
On its interval of convergence, this series converges to $\ln ( 1 + x )$. Let $f$ be the function defined by
$$f ( x ) = x \ln \left( 1 + \frac { x } { 3 } \right)$$
(a) Write the first four nonzero terms and the general term of the Maclaurin series for $f$.
(b) Determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.
(c) Let $P _ { 4 } ( x )$ be the fourth-degree Taylor polynomial for $f$ about $x = 0$. Use the alternating series error bound to find an upper bound for $\left| P _ { 4 } ( 2 ) - f ( 2 ) \right|$.