12.2.1What does it mean for a Taylor series to converge?

Interval of Convergence Using Technology

12-62.

Recall that the sum of an infinite geometric series, if it converges, can be determined by the formula $S = \frac { a } { 1 - r }$ where $a$ is the first term of the series and $r$ is the common ratio.

For each infinite geometric series below, determine the sum if it exists.
1. $S = 5 + \frac { 5 } { 3 } + \frac { 5 } { 9 } + \frac { 5 } { 27 } +\ldots$
2. $S = 100 - 50 + 25 -\frac { 25 } { 2 }+ ...$
3. $S = 2 + 5 + \frac { 25 } { 2 } + \frac { 125 } { 2 } + \frac { 625 } { 2 } +\ldots$
For what values of $r$ will an infinite geometric series converge?
Do geometric series have an open or closed interval of convergence? Explain.

12-63.

Let $f(x) =\frac { 1 } { 1 - x }$ . Notice that f can be interpreted as $S = \frac { a } { 1 - r }$ , where $a = 1$ and $r = x$ .

Work backwards. Write the geometric series whose sum is $S =\frac { 1 } { 1 - x }$ . List the first four terms and the general term.
List the first four terms and the general term of the Maclaurin series, $p(x)$ , for $f(x) =\frac { 1 } { 1 - x }$ . Compare it to the series you found in part (a).
For what values of $x$ will $p(x)$ converge? Explain.
Does the interval of convergence that you found in part (c) have an open or a closed interval? In order to answer this question, rewrite the series using sigma notation. Then test each endpoint using any of the convergence tests you learned in Chapter 10. Be prepared to share your results with the class.

12-64.

What does it mean for a Taylor series to converge?

Use a graphing calculator or the Interval of Convergence eTool (Desmos) to sketch graphs of $f(x) =\frac { 1 } { 1 - x }$ and the Maclaurin series, $p(x)$ , that you found in problem 12-63. To represent the series $p$ , use as many terms as possible. Then answer the following questions: Click in the lower right corner of the graph to view it in full-screen mode.

In part (c) of problem 12-63, you determined that the interval of convergence for $p(x)$ is $–1 < x < 1$ . Explain how the graphs of $y = f(x)$ and $y = p(x)$ can be used to visualize this interval.
What features of the graph of $f(x) =\frac { 1 } { 1 - x }$ indicate that $p(x)$ , whose center is at $x = 0$ , will have a radius of convergence of $1$ ?
In your own words, explain the relationship between an interval of convergence and a radius of convergence.

12-65.

Consider the function $g ( x ) = \frac { 1 } { 1 - ( x / 3 ) }$ , which is equivalent to $f ( \frac { x } { 3 } )$ .

Use substitution to write an equation for $q(x)$ , the Maclaurin series for $g(x)$ .
Make a prediction about the interval of convergence for $q$ . Then check your prediction by graphing both $q$ and $g$ on the same set of axes. Use as many terms for $q$ as possible.
Use substitution to justify the relationship between the interval of convergence for $p$ and the interval of convergence for $q$ .
The Maclaurin series, $m(x)$ , converges with the function h on $-\frac { 3 } { 4 }<x<\frac { 3 } { 4 }$ . If $h(x) =\frac { 1 } { 1 - U }$ , write an equation for $U$ in terms of $x$ . Confirm that your equation is correct by sketching a graph of both $h$ and $m$ on the same set of axes, using as many terms for $m$ as possible.

12-66.

In the following Math Notes box, the Maclaurin series for $f(x) =\frac { 1 } { 1 - x }$ converges on the interval $–1 < x < 1$ . Use technology to determine the interval of convergence for the Maclaurin series for $f(x) = e^x$ , $f(x) = \cos(x)$ , and $f(x) = \sin(x)$ .

12-67.

A function $f$ has derivatives of all orders for all $x$ . Some values of $f^{ }′$ are given in the table below. It is known that $f(2) = -5$ . Homework Help ✎

$x$	$2$	$2.2$	$2.4$	$2.6$	$2.8$
$f ′(x)$	$4$	$12$	$23$	$39$	$59$

Use a trapezoidal sum with four subintervals to approximate $\int _ { 2 } ^ { 2.8 } f ^ { \prime } ( x ) d x$ .
Use your approximation from part (a) to estimate the value of $f(2.8)$ . Justify your estimate with your work.
Use Euler’s Method, starting with $x = 2$ , with four steps of equal size, to approximate $f(2.8)$ . Show your work.
If $f^{\prime\prime}(2) = 32$ and $f^{\prime\prime\prime}(2) = 72$ , write a third-degree Taylor polynomial about $x = 2$ and use it to approximate $f(2.8)$ .
How close were your estimates of $f(2.8)$ in parts (b) through (d)? Explain why this happened.

12-68.

Without a calculator, use a third-degree Maclaurin polynomial to approximate $\int _ { 0 } ^ { 0.2 } \frac { \operatorname { sin } ( x ) } { x } d x$ Homework Help ✎

Compute without a calculator

12-69.

Consider the infinite series below. For each series, decide if it diverges, converges conditionally, or converges absolutely and justify your conclusion. State the tests you used.

Homework Help ✎

$\frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 11 } + \frac { 1 } { 18 } + \frac { 1 } { 27 } + \ldots$

$\frac{1}{2}+\frac{1}{9}+\frac{1}{28}+\frac{1}{65}+\frac{1}{126}+\frac{1}{217}+\ldots$

$\operatorname { ln } ( \frac { 1 } { 2 } ) + \operatorname { ln } ( \frac { 2 } { 3 } ) + \operatorname { ln } ( \frac { 3 } { 4 } ) +\ldots$

$-2 + 1 -\frac { 2 } { 3 }+\frac { 1 } { 2 }-\frac { 2 } { 5 }+\ldots$

12-70.

Use sigma notation to write the Maclaurin series, $p(x)$ , for $f(x) = \ln(x + 1)$ . Homework Help ✎

12-71.

The function $f$ has derivatives of all orders within its radius of convergence of $\frac { 1 } { 3 }$ . Its Maclaurin series is $p ( x ) = \displaystyle \sum _ { n = 0 } ^ { \infty } 3 ^ { n } x ^ { n + 2 }$ . Homework Help ✎

What are the coefficients of the first-degree and second-degree terms of $p(x)$ ? Use those coefficients to determine if $f$ has a local maximum, local minimum or neither at $x = 0$ . Justify your answer.
Expand the Maclaurin series for $f$ out to four terms to create a fifth-degree Taylor polynomial centered at $x = 0$ , $p_5(x)$ . Then write the antiderivative of $p_5(x)$ .

12-72.

The position of a particle moving in the $xy$ -plane is given by the following parametric function: $x(t) = \frac { 5 - 6 } { 3 t - 6 } ,$ $y ( t ) = \frac { 3 t } { 2 t + 6 }$ Homework Help ✎

Determine the slope of the line tangent to the path of the particle when $t = 3$ .
Write and interpret $\frac { d ^ { 2 } y } { d x ^ { 2 } }$

12-73.

Determine the interval of convergence for each of the following power series. Homework Help ✎

$\displaystyle\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( 2 n ) ! } x ^ { 2 n }$

$\displaystyle\sum _ { n = 2 } ^ { \infty } n ^ { 2 } ( x + 1 ) ^ { n }$

$\displaystyle\sum _ { n = 1 } ^ { \infty } ( \frac { n } { 2 n - 5 } ) ^ { n } x ^ { n }$

12-74.

At this point you might have noticed a relationship between power series and Taylor series. Homework Help ✎

In your own words, describe how a Power series and a Taylor series are the same and how they are different?
Describe the process you would use to determine the interval on which a power series will converge.
Describe why it might be important to find the interval of convergence of a Taylor series?

Calculus Third Edition

12.2.1What does it mean for a Taylor series to converge?

Common Maclaurin Series