Power Series

In Exercises  in Section 8.2, you were asked to find the fourth Taylor polynomial ${\mathrm{P}}_4\left(\mathrm{x}\right)$ for the specified function and the given value of ${\mathrm{x}}_0$. In Exercises  give Lagrange’s form for the remainder ${\mathrm{R}}_4\left(\mathrm{x}\right).$

$\sqrt{\mathrm{x}},1$

In Exercises 41–48 in Section 8.2, you were asked to find the

fourth Taylor polynomial $P_{4}x$ for the specified function and

the given value of $x_0$ . In Exercises 37–44 give Lagrange’s form

for the remainder $R_4\left(x\right)$.

$\ln x, 3$

In Exercises $41-48$ in Section $8.2$, you were asked to find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and

the given value of $x_0$. In Exercises $37-44$ give Lagrange’s form for the remainder $R_4\left(x\right)$.

$\sqrt[3]{x}, 1$

In Exercises 41-48 in Section 8.2, you were asked to find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$. In Exercises 37-44 give Lagrange's form for the remainder $R_4\left(x\right)$.

In Exercises $41-48$ in Section $8.2$, you were asked to find the fourth Taylor polynomial $P_4\left(x\right)$ for the specified function and the given value of $x_0$. In Exercises $37-44$ give Lagrange's form for the remainder $R_4\left(x\right)$.

$\tan x,\frac{\pi }{4}$

In Exercises $41-48$in Section $8.2$, you were asked to find the fourth Taylor polynomial $P_4\left(x\right)$for the specified function and the given value of $x_0$. In Exercises $37-44$ give Lagrange's form for the remainder $R_4\left(x\right)$.

In Exercises $49-54$ in Section $8.2$ you were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises $45-50$ find the Lagrange's form for the remainder $R_n\left(x\right)$, and show that $\underset{n\to \infty }{\lim }{R}_n\left(x\right)=0$ on the specified interval.

$\cos x,\frac{\pi }{2},\mathrm{ℝ}$

In Exercises $49-54$ in Section $8.2$ you were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises $45-50$ find the Lagrange's form for the remainder $R_n\left(x\right)$, and show that $\underset{n\to \infty }{\lim }{R}_n\left(x\right)=0$ on the specified interval.

$e^x,1,\mathrm{ℝ}$

In Exercises $45-50$ find the Lagrange’s form for the remainder $R_n\left(x\right)$, and show that $\underset{n\to \infty }{\lim }{R}_n\left(x\right)=0$ on the specified interval.

$\sin x,\pi ,\mathrm{ℝ}$

You were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises 45-50 find the Lagrange's form for the remainder $R_n\left(x\right)$, and show that $\underset{n\to \infty }{\lim }{R}_n\left(x\right)=0$on the specified interval.

$\sqrt{x}, 1, (1/2, 3/2)$

You were asked to find the Taylor series for the specified function at the given value of $x_0$. In Exercises 45-50 find the Lagrange's form for the remainder $R_n\left(x\right)$, and show that $\underset{n\to \infty }{\lim }{R}_n\left(x\right)=0$ on the specified interval.

$\ln \left(x\right), 3, (2,4)$

You were asked to find the Taylor series for the specified function at the given value of . In Exercises 45-50 find the Lagrange's form for the remainder , and show that  on the specified interval.$\sqrt[3]{x} , 1, (1/2, 3/2)$

In Exercises 49–54 in Section 8.2, you were asked to find the Taylor series for the specified function at the given value of $x_0$. In the Given exercise find the Lagrange’s form for the remainder $R_n\left(x\right)$, and show that $lim n\to \infty$ on the specified interval.

        $\sqrt[3]{x},1,(1/2,3/2)$

Find the Maclaurin series for the functions in Exercises 51–60 

by substituting into a known Maclaurin series. Also, give the 

interval of convergence for the series

$e^{x^3}$

 $e^{x^3}$

Find the Maclaurin series for the functions in Exercises 51–60 by substituting into a known Maclaurin. 

Also, give the interval of convergence for the series.

$e^{-3x^2}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

 Find the Maclaurin series for the functions in the Given Exercises by substituting it into a known Maclaurin series. Also, give the interval of convergence for the series. 

      $\frac{1}{8+x^3}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

$\frac{x}{9-x^2}$

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

Find the Maclaurin series for the functions in Exercises 51–60

by substituting into a known Maclaurin series. Also, give the

interval of convergence for the series.

Use appropriate Maclaurin series to find the first four nonzero

terms in the Maclaurin series for the product functions in

Exercises 61–66. Also, give the interval of convergence for the

series.

Use appropriate Maclaurin series to find the first four nonzero

terms in the Maclaurin series for the product functions in

Exercises 61–66. Also, give the interval of convergence for the

series.

$e^x\cos x$

Use appropriate Maclaurin series to find the first four nonzero

terms in the Maclaurin series for the product functions in

Exercises 61–66. Also, give the interval of convergence for the

series.

$e^x \ln (1+x)$

Use appropriate Maclaurin series to find the first four nonzero

terms in the Maclaurin series for the product functions in

Exercises 61–66. Also, give the interval of convergence for the

series.

$\left(\sin 2x\right)\left({\tan }^{-1}{x}^3\right)$

Use appropriate Maclaurin series to find the first four nonzero terms in the Maclaurin series for the product functions in Exercises 61–66. Also, give the interval of convergence for the series 

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$ as alternating series. Then use Theorem 7.38 to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$ of its value? In Exercises 73-76 be sure to convert to radian measure first.

$e^{-0.3}$

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$ as alternating series. Then use Theorem 7.38 to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$ of its value? In Exercises $73-76$ be sure to convert to radian measure first.

$e^{-2}$

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$ as alternating series. Then use Theorem $7.38$ to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$ of its value? In Exercises $73-76$ be sure to convert to radian measure first.

$\ln 1.3$

Use appropriate Maclaurin series to express the quantities in Exercises $67–76$ as alternating series. Then use Theorem $7.38$to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series would be needed to approximate the given quantity to within $10-6$ of its value? In Exercises $73–76$ be sure to convert to radian measure first.

${\tan }^{-1}(-0.6)$

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$as alternating series. Then use Theorem $7.38$ to approximate the value of the specified quantities to within$0.001$of their actual value. How many terms in each series would be needed to approximate the given quantity to within ${10}^{-6}$of its value? In Exercises $73-76$be sure to convert to radian measure first.

$\sin 2°$

Use appropriate Maclaurin series to express the quantities in Exercises $67-76$ as alternating series. Then use Theorem 7.38 to approximate the value of the specified quantities to within $0.001$ of their actual value. How many terms in each series Would be needed to approximate the given quantity to within ${10}^{-6}$ of its value? In Exercises $73-76$ be sure to convert to radian measure first.

$\sin 1°$

Use appropriate Maclaurin series to express the quantities in Exercises 67–76 as alternating series. Then use Theorem 7.38 to approximate the value of the specified quantities to within 0.001 of their actual value. How many terms in each series would be needed to approximate the given quantity to within 10−6 of its value? In Exercises 73–76 be sure to convert to radian measure first. 

$\cos 5°$

Use appropriate Maclaurin series to find the first four nonzero terms in the Maclaurin series for the product functions in Exercises 61-66. Also, give the interval of convergence for the series.

$\cos 10°$

(a) Use appropriate Maclaurin series to express the quantities in series form. 

(b) Use Lagrange’s form for the remainder to bound the error in using the 5th degree Maclaurin polynomial to approximate the given quantity.

 (c) Find the smallest value of n so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within ${10}^{-6}$.

$e^{0.3}$

In Exercises 77–82,

(a) Use appropriate Maclaurin series to express the quantities
in series form.

(b) Use Lagrange’s form for the remainder to bound the error in using the 5th degree Maclaurin polynomial to approximate the given quantity.

(c) Find the smallest value of n so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within

${10}^{-6}$

$e^{0.9}$

In Exercises 77–82,

(a) Use appropriate Maclaurin series to express the quantities
in series form.

(b) Use Lagrange’s form for the remainder to bound the error
in using the 5th degree Maclaurin polynomial to approximate the given quantity.

(c) Find the smallest value of  so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within ${10}^{-6}$

$ln 0.5$

In Exercises 77–82,

(a) Use appropriate Maclaurin series to express the quantities
in series form.

(b) Use Lagrange’s form for the remainder to bound the error
in using the 5th degree Maclaurin polynomial to approximate the given quantity.

(c) Find the smallest value of  so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within 

${10}^{-6}$

$ln 0.7$

In Exercises 77–82,

(a) Use appropriate Maclaurin series to express the quantities
in series form.

(b) Use Lagrange’s form for the remainder to bound the error
in using the 5th degree Maclaurin polynomial to approximate the given quantity.

(c) Find the smallest value of  so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within 

${10}^{-6}$

$\sin 1$

In Exercises 77–82,

(a) Use appropriate Maclaurin series to express the quantities
in series form.

(b) Use Lagrange’s form for the remainder to bound the error
in using the 5th degree Maclaurin polynomial to approximate the given quantity.

(c) Find the smallest value of  so that the nth degree Maclaurin polynomial approximation to the given quantity is guaranteed to be accurate to within 

${10}^{-6}$

$\cos 2$

Let, $f\left(x\right)=\left\{\begin{array}{l}\frac{\sin x}{x}, if x\ne 0\\ 0 , if x=0\end{array}\right.$

(a) Use the definition of the derivative to prove that $f$ is differentiable at $0$

(b) Use the Maclaurin series for $\sin x$ to find a Maclaurin
series for $f$

Let,

$f\left(x\right)=\left\{\begin{array}{l}\frac{1-\cos x}{x} ,if x\ne 0\\ 0 ,if x=0 \end{array}\right.$

(a) Use the definition of the derivative to prove that $f$ is differentiable at $0$

(b) Use the Maclaurin series for $\cos x$ to find a Maclaurin series for $f$

Emmy is a civil engineer who has to approximate the slope of the water table along a certain line in the Hanford Nuclear Reservation. She can dig only three test holes to evaluate the depth of groundwater at certain points.
She finds that the water table lies at $(0, 35)$, $(300, 38)$, and $(600, 42)$, with all distances given in feet and positive vertical distances representing distances below the surface.

(a) Using the data from the first two wells, estimate the slope of the water table. Use this slope to write an equation of a line that describes the water table.

(b) Considering the equation of the line from part (a) as the first two terms of a Maclaurin series for the function $w\left(x\right)$ describing the water table, write an
expression for the error in this linear approximation.

(c) Verify that the graph of the quadratic

 $w_2\left(x\right)=\frac{1}{180000}{x}^2+\frac{1}{120}x+35$

passes through the three data points describing the
water table.

(c) Use the quadratic from part (c) to estimate the error
in the linear approximation to the water table.

Use Lagrange’s form for the remainder to prove that the

Maclaurin series for the cosine,

$\cos x=\sum _{k=0}^{\mathrm{\infty }} \frac{(-1{)}^k}{\left(2k\right)!}{x}^{2k}$

converges for every real number.

Use the Maclaurin series for $e^x$ and $e^{-x}$ to prove that

$\sinh x=\sum _{k=0}^{\infty }\frac{1}{(2k+1)!}{x}^{2k+1}.$

If $\mathrm{f}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{\infty }} {\mathrm{a}}_{\mathrm{k}}{\left(\mathrm{x}-{\mathrm{x}}_0\right)}^{\mathrm{k}}$ , what is $\mathrm{f}\left({\mathrm{x}}_0\right)?$What is $\mathrm{f}\text{'}\left({\mathrm{x}}_0\right)?$ What is $\mathrm{f}"\left({\mathrm{x}}_0\right)?$What is ${\mathrm{f}}^{\left(\mathrm{k}\right)}\left({\mathrm{x}}_0\right)?$

Let $f\left(x\right)=\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k$ and let $G$ be the antiderivative for $f$ with the property that $G\left(x_0\right)=7$. Find the Taylor series in$x_0$ for $G$.

 Let $f(x$) be a function such that the power series in $x-x_0,$ $\sum _{k-0}^{\infty }{a}_k{\left(x-x_0\right)}^k$ converges absolutely to $f$ on the interval I. If $G_1$ and $G_2$ are two antiderivatives for $f$, explain why the power series in $x-x_0$ for $G_1$and $G_2$ have the same interval of convergence.

In Example 1 we used Theorem 8.11 to find the Maclaurin series for $\frac{1}{(1-x{)}^2}.$ Explain how Theorem 8.11 could be used to find the Maclaurin series for $\frac{1}{(1-x{)}^2}$, where $k$ is a positive integer greater than $2$.