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Chapter 7: Q. 2 TF (page 633)

Find all values of x for which the series ${鈭�}_{k = 1}^{鈭�} \frac{|x|}{4^{k}}$ converges.

Short Answer

Expert verified

So series cannot converse for any value of x.

Step by step solution

01

Step 1. Given information.

The given series is ${鈭�}_{k = 0}^{鈭�} \frac{|x|}{4^{k}} .$

02

Step 2. Value of x.

Use the root test and Find the value of $蚁 = \lim_{k 鈫� 鈭�} {(a_{k})}^{1 / k}$

role="math" localid="1649198960481" $蚁 = \lim_{k 鈫� 鈭�} {(a_{k})}^{1 / k} 蚁 = \lim_{k 鈫� 鈭�} {(\frac{|x|}{4^{k}})}^{\frac{1}{k}} 蚁 = 4 路 \lim_{k 鈫� 鈭�} {|x|}^{\frac{1}{k}} 蚁 = 4 路 \lim_{k 鈫� 鈭�} e^{\frac{1}{k} \ln |x|} 蚁 = 4 路 {(\lim_{k 鈫� 鈭�} e^{\frac{1}{k}})}^{\ln | x |} 蚁 = 4 (1^{\ln | x |})$

According to the root test, series will converge when $蚁 < 1 .$

$4 (1^{\ln | x |}) < 1$

there is no solution for $x 鈭� R .$

So series cannot converse for any value ofx.

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Most popular questions from this chapter

Find the values of x for which the series ${鈭�}_{k = 0}^{鈭�} x^{k}$ converges.

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{k^{k}}$

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Explain how you could adapt the integral test to analyze a series ${鈭�}_{k = 1}^{鈭�} f (k)$ in which the function $f : [1, 鈭�) 鈫� 鈩�$ is continuous, negative, and increasing.

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