A function $f$ has derivatives of all orders for $-1 < x < 1$. The derivatives of $f$ satisfy the conditions below. The Maclaurin series for $f$ converges to $f(x)$ for $|x| < 1$. $$\begin{aligned}
f(0) &= 0 \\
f'(0) &= 1 \\
f^{(n+1)}(0) &= -n \cdot f^{(n)}(0) \text{ for all } n \geq 1
\end{aligned}$$ (a) Show that the first four nonzero terms of the Maclaurin series for $f$ are $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$, and write the general term of the Maclaurin series for $f$. (b) Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at $x = 1$. Explain your reasoning. (c) Write the first four nonzero terms and the general term of the Maclaurin series for $g(x) = \int_{0}^{x} f(t)\, dt$. (d) Let $P_n\!\left(\frac{1}{2}\right)$ represent the $n$th-degree Taylor polynomial for $g$ about $x = 0$ evaluated at $x = \frac{1}{2}$, where $g$ is the function defined in part (c). Use the alternating series error bound to show that $\left|P_4\!\left(\frac{1}{2}\right) - g\!\left(\frac{1}{2}\right)\right| < \frac{1}{500}$.
A function $f$ has derivatives of all orders for $-1 < x < 1$. The derivatives of $f$ satisfy the conditions below. The Maclaurin series for $f$ converges to $f(x)$ for $|x| < 1$.
$$\begin{aligned}
f(0) &= 0 \\
f'(0) &= 1 \\
f^{(n+1)}(0) &= -n \cdot f^{(n)}(0) \text{ for all } n \geq 1
\end{aligned}$$
(a) Show that the first four nonzero terms of the Maclaurin series for $f$ are $x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}$, and write the general term of the Maclaurin series for $f$.\\
(b) Determine whether the Maclaurin series described in part (a) converges absolutely, converges conditionally, or diverges at $x = 1$. Explain your reasoning.\\
(c) Write the first four nonzero terms and the general term of the Maclaurin series for $g(x) = \int_{0}^{x} f(t)\, dt$.\\
(d) Let $P_n\!\left(\frac{1}{2}\right)$ represent the $n$th-degree Taylor polynomial for $g$ about $x = 0$ evaluated at $x = \frac{1}{2}$, where $g$ is the function defined in part (c). Use the alternating series error bound to show that $\left|P_4\!\left(\frac{1}{2}\right) - g\!\left(\frac{1}{2}\right)\right| < \frac{1}{500}$.