The function $f$ is defined by the power series $$f(x) = -\frac{x}{2} + \frac{2x^{2}}{3} - \frac{3x^{3}}{4} + \cdots + \frac{(-1)^{n} n x^{n}}{n+1} + \cdots$$ for all real numbers $x$ for which the series converges. The function $g$ is defined by the power series $$g(x) = 1 - \frac{x}{2!} + \frac{x^{2}}{4!} - \frac{x^{3}}{6!} + \cdots + \frac{(-1)^{n} x^{n}}{(2n)!} + \cdots$$ for all real numbers $x$ for which the series converges.
(a) Find the interval of convergence of the power series for $f$. Justify your answer.
(b) The graph of $y = f(x) - g(x)$ passes through the point $(0, -1)$. Find $y'(0)$ and $y''(0)$. Determine whether $y$ has a relative minimum, a relative maximum, or neither at $x = 0$. Give a reason for your answer.

The function $f$, defined by $$f(x) = \begin{cases} \frac{\cos x - 1}{x^2} & \text{for } x \neq 0 \\ -\frac{1}{2} & \text{for } x = 0 \end{cases}$$ has derivatives of all orders. Let $g$ be the function defined by $g(x) = 1 + \int_{0}^{x} f(t)\,dt$.
(a) Write the first three nonzero terms and the general term of the Taylor series for $\cos x$ about $x = 0$. Use this series to write the first three nonzero terms and the general term of the Taylor series for $f$ about $x = 0$.
(b) Use the Taylor series for $f$ about $x = 0$ found in part (a) to determine whether $f$ has a relative maximum, relative minimum, or neither at $x = 0$. Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for $g$ about $x = 0$.
(d) The Taylor series for $g$ about $x = 0$, evaluated at $x = 1$, is an alternating series with individual terms that decrease in absolute value to 0. Use the third-degree Taylor polynomial for $g$ about $x = 0$ to estimate the value of $g(1)$. Explain why this estimate differs from the actual value of $g(1)$ by less than $\frac{1}{6!}$.