The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is $$\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{2n+3} = \frac{x}{3} - \frac{x^{3}}{5} + \frac{x^{5}}{7} - \cdots$$ (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 $g'(x)$.
The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum_{n=0}^{\infty} (-1)^{n} \frac{x^{2n+1}}{2n+3} = \frac{x}{3} - \frac{x^{3}}{5} + \frac{x^{5}}{7} - \cdots$$
(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 $g'(x)$.