Power Series

$Let f\left(x\right) = \infty$$k=0 ak(x - x 0)$k and let G be an antiderivative

for f . Explain why we do not have enough information to

determine $G(x 0).$ What is G $(x 0) What is G$$(x 0) What is G\left(k\right) (x 0)$?

Let$f\left(x\right)=\sum _{k=0}^{\infty }{a}_k{\left(x-x_0\right)}^k$and let G be an antiderivative for f. Explain why we do not have enough information to determine $G\left(x_0\right).What is G^{\text{'}}\left(x_0\right)?$ What is $G^{\text{'}\text{'}}\left(x_0\right)?$ What is $G^{\left(k\right)}\left(x_0\right)?$

If $f$ is a function such that $f\left(0\right)=1$ and $f^{\text{'}}\left(x\right)=f\left(x\right)$ for every value of $x$, find the Maclaurin series for $f$.

If $f$ is a function such that $f\left(0\right) = -3$ and $f\text{'}\left(x\right) = 2f\left(x\right)$ every value of $x$, find the Maclaurin series for $f$.

Perform the following steps for the power series in $x - x_0$ in Exercises 11–16:

(a) Find the interval of convergence, $I$, for the series.

(b) Let $f$ be the function to which the series converges on $I.$ Find the power series in $x - x_0$ for $f$

(c) Find the power series in $x-x_0$ for$F\left(x\right)={\int }_{x_0}^{x}f\left(t\right)dt$

11. $\sum _{k=0}^{\infty }{2}^k{x}^k$

Perform the following steps for the power series in$x-x_0$ in Exercises 11 -16

(a) Find the interval of convergence, $I$, for the series.

(b) Let $f$ be the function to which the series converges on$I$. Find the power series in $x-x_0$for $f^{\text{'}}.$

(c) Find the power series in $x-x_0$ for$F\left(x\right)={\int }_{x_0}^{x}f\left(t\right)dt$

12. $\sum _{k=0}^{\infty }\frac{5^k}{k!}(x-3{)}^k$

Perform the following steps for the power series in x − x 0 in

Exercises 11–16:

(a) Find the interval of covergence, I, for the series.

(b) Let f be the function to which the series converges on I.

Find the power series in x − x 0 for f 

(c) Find the power series in uncaught exception: Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('e70721f5d251e47...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500') #1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500') #2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500') #3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500') #4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500') #5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL) #6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true) #7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true) #8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('e70721f5d251e47...', NULL, Object(PhpParamsProvider)) #9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

x 0 f(t) dt

$\sum _{k=0}^{\infty }\frac{1}{k+2}{x}^{3k+1}$

 Perform the following steps for the power series in $x-x_0$ in the given exercises:

(a)  Find the interval of convergence, I, for the series. 

(b)  Let f be the function to which the series converges on I. Find the power series in $x-x_0$ for f.

(c)  Find the power series in $x-x_0\text{ for }F\left(x\right)={\int }_{x_0}^x f\left(t\right)dt$.

             $\sum _{k=0}^{\mathrm{\infty }} (-1{)}^k\frac{k!}{\left(2k\right)!}(x-7{)}^{2k}$

In exercise 26-30 Find a definite integral that represents the length of the specified polar curve and then find the exact value of integral  

The cardoid $r=2-2\sin 2\theta for 0\le \theta \le 2\pi$

Find Maclaurin series for the given pairs of functions, using these steps: (a) Use substitution in the appropriate Maclaurin series to find the Maclaurin series for the given function. (b) Use Theorem 8.11 and your answer from part (a) to find the Maclaurin series for the given function. (c) Find the Maclaurin series for the function in (b), using multiplication and substitution with the appropriate Maclaurin series. Compare your answers from (b) and (c).

(a) $\cos \left(4x^3\right)$ 

(b) $x^2\sin \left(4x^3\right)$

Find Maclaurin series for the given pairs of functions, using these steps: (a) Use substitution in the appropriate Maclaurin series to find the Maclaurin series for the given function. (b) Use Theorem 8.11 and your answer from part (a) to find the Maclaurin series for the given function. (c) Find the Maclaurin series for the function in (b), using multiplication and substitution with the appropriate Maclaurin series. Compare your answers from (b) and (c).

(a) $e^{-x^2}$

(b) $x{e}^{-x^2}$

In Exercises 31–40 find the Maclaurin series for the specified function. Note: These are the same functions as in Exercises 21–30.

$f\left(x\right)=\ln (1+x)$

Find Maclaurin series for the given pairs of functions, using these steps: (a) Use substitution in the appropriate Maclaurin series to find the Maclaurin series for the given function. (b) Use Theorem 8.11 and your answer from part (a) to find the Maclaurin series for the given function. (c) Find the Maclaurin series for the function in (b), using multiplication and substitution with the appropriate Maclaurin series. Compare your answers from (b) and (c). 

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

(b) $\left(\frac{x}{9+x^4}\right)$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) $\frac{1}{1-x}$, $x_0=3$

(b) $\frac{1}{{\left(1-x\right)}^2}$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) $\frac{1}{1-x}$, $x_0=-3$

(b) $\frac{1}{{\left(1-x\right)}^2}$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) $\frac{1}{2-x}$, $x_0=3$

(b) $\frac{1}{{\left(2-x\right)}^2}$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) $\frac{1}{2-x}$, $x_0=-3$

(b) $\frac{1}{{\left(2-x\right)}^2}$

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) ,

(b)

Explore the Taylor series for the given pairs of functions, using these steps: (a) Find the Taylor series for the given function at the specified value of x 0 and determine the interval of convergence for the series. (b) Use Theorem 8.11 and your answer from part (a) to find the Taylor series for the given function for the same value of x 0. Also, find the interval of convergence for your series.

(a) ,

(b)

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 $\frac{{\tan }^{-1}x}{1-x^3}$. 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.

$\ln 1.5$

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.

${\tan }^{-1}0.4$

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}$$73-76$ of its value? In Exercises  be sure to convert to radian measure first.

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

What is the definition of an odd function? An even function?

What is a power series in x?

Fill in the blanks: The graph of every odd function is symmetric about ______. The graph of every even function is symmetric about ______.

What is a power series in $x-x_0$?

Explain why $\sum _{k=0}^{\infty }{\left(x-k\right)}^k$ is not a power series.

What is meant by the interval of convergence for a power series in $x$? How is the interval of convergence determined? If a power series in $x$ has a nontrivial interval of convergence, what types of intervals are possible?

What is meant by the interval of convergence for a power series in $x-x_0$? How is the interval of convergence determined? If a power series in $x-x_0$ has a nontrivial interval of convergence, what types of intervals are possible. 

Show that $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{1+2^k}{x}^k$, the power series in $x$ from Example 1, diverges when  $x=-2$

Complete Example 4 by showing that the power series $\sum _{k=0}^{\mathrm{\infty }} (-1{)}^k\frac{k}{3k+1}(2x-3{)}^k$ diverges when $x=2$.

Show that the power series $\sum _{k=0}^{\mathrm{\infty }} \frac{(-1{)}^k}{2k+1}{x}^{2k+1}$ converges conditionally when $x=1$ and when $x=-1$. What does this behavior tell you about the interval of convergence for the series?

Show that the power series $\sum _{k=1}^{\mathrm{\infty }} \frac{(-1{)}^k}{k}{x}^k$ converges conditionally when $x=1$ and diverges when $x=-1$. What does this behavior tell you about the interval of convergence for the series?

Show that the power series $\sum _{k=1}^{\mathrm{\infty }} \frac{(-1{)}^k}{k^2}{x}^k$ converges absolutely when $x=1$ and when $x=-1$. What does this behavior tell you about the interval of convergence for the series?

What is $x_0$ if the interval of convergence for the power series $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k\text{ is }(2,10]?$

What is $x_0$ if $(p,q)$ is the interval of convergence for the power series $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$ ?

What is $x_0$ if the power series $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$ converges conditionally at both $x=-4$ and $x=8$.

Is it possible for a power series to have $(0,\infty )$ as its interval converge? Explain your answer.

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in $x$ with a positive and finite radius of convergence $p$. Explain why the ratio test for absolute convergence will fail to determine the convergence of this power series when $x=p$ or when $x=-p$.

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in x with a radius of convergence $p$. What is the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} a_k{\left(x-x_0\right)}^k$? Make sure you justify your answer.

Let $a_k\ne 0$for each value of $k$, and let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in $x$ with a positive and finite radius of convergence $p$. What is the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} \frac{1}{a_k}{x}^k$ ?

Let $\sum _{k=0}^{\mathrm{\infty }} a_k{x}^k$ be a power series in x with an interval of convergence$[-2,2)$. What is the radius of convergence of the power series $\sum _{k=0}^{\mathrm{\infty }} a_k(x-3{)}^k$? Justify your answer. 

Find the interval of convergence for power series: $\sum _{k=0}^{\infty }\frac{1}{k! }{x}^k$

Find the interval of convergence for power series: $\sum _{k=0}^{\infty }\frac{{\left(-2\right)}^k}{k!}{x}^k$

Find the interval of convergence for power series: $\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k}{\left(2k\right)!}{x}^{2k}$.

Find the interval of convergence for power series: $\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^k}{\left(2k+1\right)!}{x}^{2k+1}$