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Chapter 8: Q 8. (page 704)

Find the interval of convergence of the power series
${鈭�}_{k = 1}^{鈭�} (x - 3)^{k}$

Short Answer

Expert verified

The interval of convergence of the power series ${鈭�}_{k = 1}^{鈭�} (x - 3)^{k}$ is $(2, 4)$

Step by step solution

01

Given information

The power series is ${鈭�}_{k = 1}^{鈭�} (x - 3)^{k}$

02

The ratio test for absolute convergence will be used to determine the convergence interval.

Let, the first assume, therefore

\lim_{k 鈫� 鈭�} |\frac{b_{k + 1}}{b_{k}}| = \lim_{k 鈫� 鈭�} |\frac{{(x - 3)}^{k + 1}}{{(x - 3)}^{k}}| = \lim_{k 鈫� 鈭�} |x - 3|

The limit is $| x - 3 |$

The ratio test for absolute convergence will be used to determine the convergence interval when $| x - 3 | < 1$ that is $- 1 < x - 3 < 1$

As a result, we may write $- 1 < x - 3$ and $x - 3 < 1$

Implies that

$x > 2 a n d x < 4$

or $x 鈭� (2, 4)$

03

Now, because the intervals are limited, we examine the series' behavior at the ends.

When $x = 2$

${{鈭�}_{k = 1}^{鈭�} (x - 3)^{k}|}_{x = 2} = {鈭�}_{k = 1}^{鈭�} (2 - 3)^{k}$

$= {鈭�}_{k = 1}^{鈭�} (- 1)^{k}$

The series contains the conditionally convergent alternating multiple.

When $x = 4$

${{鈭�}_{k = 1}^{鈭�} (x - 3)^{k}|}_{x = 4} = {鈭�}_{k = 1}^{鈭�} (4 - 3)^{k} = {鈭�}_{k = 1}^{鈭�} (1)^{k} = {鈭�}_{k = 1}^{鈭�} 1$

The series is a diverging constant multiple.

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Most popular questions from this chapter

Given a function f and a Taylor polynomial for fat $x_{0}$ , what is meant by the nth remainder $R_{n} (x)$ ? What does $R_{n} (x)$ measure?

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

${鈭�}_{k = 1}^{鈭�} \frac{2^{k} + 1}{\sqrt{k}} x^{k}$

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

${鈭�}_{k = 0}^{鈭�} \frac{1}{3^{k} + 5^{k}} x^{k}$

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

In Exercises 41鈥�48 find the fourth Taylor polynomial $P_{4} (x)$ for the specified function and the given value of $x_{0}$ .

$48. \sin 3 x, \frac{蟺}{6}$

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