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 $g'(x)$.

For $x > 0$, the power series $1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$ converges to which of the following?
(A) $\cos x$
(B) $\sin x$
(C) $\frac { \sin x } { x }$
(D) $e ^ { x } - e ^ { x ^ { 2 } }$
(E) $1 + e ^ { x } - e ^ { x ^ { 2 } }$

Let $f$ be a function having derivatives of all orders for $x > 0$ such that $f ( 3 ) = 2 , f ^ { \prime } ( 3 ) = - 1 , f ^ { \prime \prime } ( 3 ) = 6$, and $f ^ { \prime \prime \prime } ( 3 ) = 12$. Which of the following is the third-degree Taylor polynomial for $f$ about $x = 3$ ?
(A) $2 - x + 6 x ^ { 2 } + 12 x ^ { 3 }$
(B) $2 - x + 3 x ^ { 2 } + 2 x ^ { 3 }$
(C) $2 - ( x - 3 ) + 6 ( x - 3 ) ^ { 2 } + 12 ( x - 3 ) ^ { 3 }$
(D) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 4 ( x - 3 ) ^ { 3 }$
(E) $2 - ( x - 3 ) + 3 ( x - 3 ) ^ { 2 } + 2 ( x - 3 ) ^ { 3 }$

A function $f$ has derivatives of all orders at $x = 0$. Let $P _ { n } ( x )$ denote the $n$th-degree Taylor polynomial for $f$ about $x = 0$.
(a) It is known that $f ( 0 ) = - 4$ and that $P _ { 1 } \left( \frac { 1 } { 2 } \right) = - 3$. Show that $f ^ { \prime } ( 0 ) = 2$.
(b) It is known that $f ^ { \prime \prime } ( 0 ) = - \frac { 2 } { 3 }$ and $f ^ { \prime \prime \prime } ( 0 ) = \frac { 1 } { 3 }$. Find $P _ { 3 } ( x )$.
(c) The function $h$ has first derivative given by $h ^ { \prime } ( x ) = f ( 2 x )$. It is known that $h ( 0 ) = 7$. Find the third-degree Taylor polynomial for $h$ about $x = 0$.

The Taylor series for a function $f$ about $x = 1$ is given by $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } \frac { 2 ^ { n } } { n } ( x - 1 ) ^ { n }$ and converges to $f ( x )$ for $| x - 1 | < R$, where $R$ is the radius of convergence of the Taylor series.
(a) Find the value of $R$.
(b) Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 1$.
(c) The Taylor series for $f ^ { \prime }$ about $x = 1$, found in part (b), is a geometric series. Find the function $f ^ { \prime }$ to which the series converges for $| x - 1 | < R$. Use this function to determine $f$ for $| x - 1 | < R$.

The Maclaurin series for a function $f$ is given by $$\sum _ { n = 1 } ^ { \infty } \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } = x - \frac { 3 } { 2 } x ^ { 2 } + 3 x ^ { 3 } - \cdots + \frac { ( - 3 ) ^ { n - 1 } } { n } x ^ { n } + \cdots$$ and converges to $f ( x )$ for $| x | < R$, where $R$ is the radius of convergence of the Maclaurin series.
(a) Use the ratio test to find $R$.
(b) Write the first four nonzero terms of the Maclaurin series for $f ^ { \prime }$, the derivative of $f$. Express $f ^ { \prime }$ as a rational function for $| x | < R$.
(c) Write the first four nonzero terms of the Maclaurin series for $e ^ { x }$. Use the Maclaurin series for $e ^ { x }$ to write the third-degree Taylor polynomial for $g ( x ) = e ^ { x } f ( x )$ about $x = 0$.

The function $f$ has a Taylor series about $x = 1$ that converges to $f ( x )$ for all $x$ in the interval of convergence. It is known that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - \frac { 1 } { 2 }$, and the $n$th derivative of $f$ at $x = 1$ is given by $f ^ { ( n ) } ( 1 ) = ( - 1 ) ^ { n } \frac { ( n - 1 ) ! } { 2 ^ { n } }$ for $n \geq 2$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$.
(b) The Taylor series for $f$ about $x = 1$ has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer.
(c) The Taylor series for $f$ about $x = 1$ can be used to represent $f ( 1.2 )$ as an alternating series. Use the first three nonzero terms of the alternating series to approximate $f ( 1.2 )$.
(d) Show that the approximation found in part (c) is within 0.001 of the exact value of $f ( 1.2 )$.

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 $\left|P_4\!\left(\frac{1}{2}\right) - g\!\left(\frac{1}{2}\right)\right| < \frac{1}{500}$.

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 $\left| P _ { 4 } ( 2 ) - f ( 2 ) \right|$.

A function $f$ has derivatives of all orders for all real numbers $x$. A portion of the graph of $f$ is shown above, along with the line tangent to the graph of $f$ at $x = 0$. Selected derivatives of $f$ at $x = 0$ are given in the table below.

$n$	$f ^ { ( n ) } ( 0 )$
2	3
3	$-\frac { 23 } { 2 }$
4	54

(a) Write the third-degree Taylor polynomial for $f$ about $x = 0$.
(b) Write the first three nonzero terms of the Maclaurin series for $e ^ { x }$. Write the second-degree Taylor polynomial for $e ^ { x } f ( x )$ about $x = 0$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, dt$. Use the Taylor polynomial found in part (a) to find an approximation for $h ( 1 )$.
(d) It is known that the Maclaurin series for $h$ converges to $h ( x )$ for all real numbers $x$. It is also known that the individual terms of the series for $h ( 1 )$ alternate in sign and decrease in absolute value to 0. Use the alternating series error bound to show that the approximation found in part (c) differs from $h ( 1 )$ by at most 0.45.

Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$. It can be shown that $f ^ { \prime \prime } ( 1 ) = 4$.
(a) Write the second-degree Taylor polynomial for $f$ about $x = 1$. Use the Taylor polynomial to approximate $f ( 2 )$.
(b) Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 2 )$. Show the work that leads to your answer.
(c) Find the particular solution $y = f ( x )$ to the differential equation $\frac { d y } { d x } = y \cdot ( x \ln x )$ with initial condition $f ( 1 ) = 4$.

The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 2 e ^ { n } + 3 }$ on its interval of convergence.
(a) State the conditions necessary to use the integral test to determine convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$. Use the integral test to show that $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ converges.
(b) Use the limit comparison test with the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ to show that the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ converges absolutely.
(c) Determine the radius of convergence of the Maclaurin series for $g$.
(d) The first two terms of the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ are used to approximate $g ( 1 )$. Use the alternating series error bound to determine an upper bound on the error of the approximation.

The function $f$ is defined by the power series $f ( x ) = x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 } - \frac { x ^ { 7 } } { 7 } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n + 1 } } { 2 n + 1 } + \cdots$ for all real numbers $x$ for which the series converges.
(a) Using the ratio test, find the interval of convergence of the power series for $f$. Justify your answer.
(b) Show that $\left| f \left( \frac { 1 } { 2 } \right) - \frac { 1 } { 2 } \right| < \frac { 1 } { 10 }$. Justify your answer.
(c) Write the first four nonzero terms and the general term for an infinite series that represents $f ^ { \prime } ( x )$.
(d) Use the result from part (c) to find the value of $f ^ { \prime } \left( \frac { 1 } { 6 } \right)$.

The function $f$ has derivatives of all orders for all real numbers. It is known that $f(0) = 2$, $f'(0) = 3$, $f''(x) = -f\left(x^{2}\right)$, and $f'''(x) = -2x \cdot f'\left(x^{2}\right)$.
(a) Find $f^{(4)}(x)$, the fourth derivative of $f$ with respect to $x$. Write the fourth-degree Taylor polynomial for $f$ about $x = 0$. Show the work that leads to your answer.
(b) The fourth-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f(0.1)$. Given that $\left|f^{(5)}(x)\right| \leq 15$ for $0 \leq x \leq 0.5$, use the Lagrange error bound to show that this approximation is within $\frac{1}{10^{5}}$ of the exact value of $f(0.1)$.
(c) Let $g$ be the function such that $g(0) = 4$ and $g'(x) = e^{x} f(x)$. Write the second-degree Taylor polynomial for $g$ about $x = 0$.

The function $f$ is twice differentiable for all $x$ with $f(0) = 0$. Values of $f'$, the derivative of $f$, are given in the table for selected values of $x$.

$x$	0	$\pi$	$2\pi$
$f'(x)$	5	6	0

(a) For $x \geq 0$, the function $h$ is defined by $h(x) = \int_{0}^{x} \sqrt{1 + \left(f'(t)\right)^2}\, dt$. Find the value of $h'(\pi)$. Show the work that leads to your answer.
(b) What information does $\int_{0}^{\pi} \sqrt{1 + \left(f'(x)\right)^2}\, dx$ provide about the graph of $f$?
(c) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f(2\pi)$. Show the computations that lead to your answer.
(d) Find $\int (t + 5)\cos\left(\frac{t}{4}\right)\, dt$. Show the work that leads to your answer.

The Maclaurin series for a function $f$ is given by $\sum_{n=1}^{\infty} \frac{(n+1)x^n}{n^2 6^n}$ and converges to $f(x)$ for all $x$ in the interval of convergence. It can be shown that the Maclaurin series for $f$ has a radius of convergence of 6.
(a) Determine whether the Maclaurin series for $f$ converges or diverges at $x = 6$. Give a reason for your answer.
(b) It can be shown that $f(-3) = \sum_{n=1}^{\infty} \frac{(n+1)(-3)^n}{n^2 6^n} = \sum_{n=1}^{\infty} \frac{n+1}{n^2}\left(-\frac{1}{2}\right)^n$ and that the first three terms of this series sum to $S_3 = -\frac{125}{144}$. Show that $\left|f(-3) - S_3\right| < \frac{1}{50}$.
(c) Find the general term of the Maclaurin series for $f'$, the derivative of $f$. Find the radius of convergence of the Maclaurin series for $f'$.
(d) Let $g(x) = \sum_{n=1}^{\infty} \frac{(n+1)x^{2n}}{n^2 3^n}$. Use the ratio test to determine the radius of convergence of the Maclaurin series for $g$.

The Taylor series for a function $f$ about $x = 4$ is given by $$\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n + 1 } } { ( n + 1 ) 3 ^ { n } } = \frac { ( x - 4 ) ^ { 2 } } { 2 \cdot 3 } + \frac { ( x - 4 ) ^ { 3 } } { 3 \cdot 3 ^ { 2 } } + \frac { ( x - 4 ) ^ { 4 } } { 4 \cdot 3 ^ { 3 } } + \cdots + \frac { ( x - 4 ) ^ { n + 1 } } { ( n + 1 ) 3 ^ { n } } + \cdots$$ and converges to $f ( x )$ on its interval of convergence.
A. Using the ratio test, find the interval of convergence of the Taylor series for $f$ about $x = 4$. Justify your answer.
B. Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 4$.
C. The Taylor series for $f ^ { \prime }$ described in part B is a geometric series. For all $x$ in the interval of convergence of the Taylor series for $f ^ { \prime }$, show that $f ^ { \prime } ( x ) = \frac { x - 4 } { 7 - x }$.
D. It is known that the radius of convergence of the Taylor series for $f$ about $x = 4$ is the same as the radius of convergence of the Taylor series for $f ^ { \prime }$ about $x = 4$. Does the Taylor series for $f ^ { \prime }$ described in part B converge to $f ^ { \prime } ( x ) = \frac { x - 4 } { 7 - x }$ at $x = 8$ ? Give a reason for your answer.

The power series $$\sum_{n=1}^{\infty} \frac{n^2 x^n}{n!}$$ equals
(A) $x^2 e^x$;
(B) $x e^x$;
(C) $(x^2 + x) e^x$;
(D) $(x^2 - x) e^x$;

$$f ( x ) = \frac { x } { x + \sin x } \quad \text { and } \quad g ( x ) = \frac { x ^ { 4 } + x ^ { 6 } } { e ^ { x } - 1 - x ^ { 2 } }$$
(a) Limit as $x \rightarrow 0$ of $f ( x )$ is $\frac { 1 } { 2 }$.
(b) Limit as $x \rightarrow \infty$ of $f ( x )$ does not exist.
(c) Limit as $x \rightarrow \infty$ of $g ( x )$ is finite.
(d) Limit as $x \rightarrow 0$ of $g ( x )$ is 720.

Find the Taylor series expansion of $f ( x ) = e ^ { x } \sin ( x )$ centered at $x = \frac { \pi } { 4 }$.

Find the Taylor series expansion of $f ( x ) = e ^ { x } \sin ( x )$ centered at $x = \frac { \pi } { 4 }$.

Give the power series expansion of the function $t \mapsto \frac { 1 } { ( 1 - t ) ^ { 2 } }$ as well as its radius of convergence. Specify whether the series converges at the endpoints of the interval of convergence.

We consider the functions $\varphi : x \mapsto \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ and $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$. Determine sequences $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ such that, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { n = 0 } ^ { + \infty } u _ { n } x ^ { n } \quad \text { and } \quad \psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n } .$$ We will express explicitly as a function of $n$, according to the parity of $n$, the reals $u _ { n }$ and $v _ { n }$.

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $A_n = \sum_{k=0}^n a_k$ and $\widetilde{a}_n = \frac{A_n}{n+1}$.
Calculate $\widetilde { v } _ { n }$ (arithmetic mean of the numbers $v _ { 0 } , \ldots , v _ { n }$).

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $\widetilde{a}_n = \frac{1}{n+1}\sum_{k=0}^n a_k$.
Construct using $\psi$ an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not satisfying property II.3 ($\lim_{n\to\infty} \widetilde{a}_n = 1$).