The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is $$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$ (a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$. (b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$. (c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.
The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$.
(b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$.
(c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.