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Chapter 7: Q. 38 (page 657)

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.
${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{k}{4}$ ^k

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{k}{4^{k}}$ converges absolutely.

Step by step solution

01

Step 1. Given Information.

The series:

${鈭�}_{k = 1}^{鈭�} {(- 1)}^{k} \frac{k}{4^{k}}$

02

Step 2. By Alternating Series Test.

According to the Alternating Series Test, the sequence $a_{k + 1} < a_{k}$ for every . Then the alternating series $a_{k + 1}, a_{k}$ both converges.

03

Step 3. Find ak+1.

$a_{k} = \frac{k}{4^{k}} a_{k + 1} = \frac{k + 1}{4^{k + 1}} a_{k + 1} < a_{k}$

So the sequence is monotonic decreasing sequence.

04

Step 4. Find limk→∞ak.

$\underset{k 鈫� 鈭�}{l i m} a_{k} = \underset{k 鈫� 鈭�}{l i m} \frac{k}{4^{k}} = \frac{鈭�}{鈭�}$

By using L'Hospital's rule,

$\underset{k 鈫� 鈭�}{l i m} a_{k} = \underset{k 鈫� 鈭�}{l i m} \frac{1}{4^{k} \ln 4} = \underset{k 鈫� 鈭�}{l i m} \frac{1}{2^{2 k} 2 \ln 2} = 0$

So the series converges.

05

Step 5. Use Ratio test.

$\underset{k 鈫� 鈭�}{l i m} |\frac{a_{k + 1}}{a_{k}}| = \underset{k 鈫� 鈭�}{l i m} |{(- 1)}^{k + 1} \frac{k + 1}{4}|$ ^k+1(-1)kk4^k=limk鈫�鈭�(k+1)4kk(4k+1)=limk鈫�鈭�k+14k=14<1

So the series converges absolutely.

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Most popular questions from this chapter

Given that $a_{0} = - 3, a_{1} = 5, a_{2} = - 4, a_{3} = 2$ and ${鈭� a_{k}}_{k = 2}^{鈭�} = 7$ , find the value ofrole="math" localid="1648828803227" ${鈭� a_{k}}_{k = 3}^{鈭�}$ .

Consider the series

Fill in the blanks and select the correct word:

$I f \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� and {鈭�}_{k = 1}^{鈭�}_____diverges t he n {鈭�}_{k = 1}^{鈭�}_____(converges / diverges) .$

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. Prove that ${鈭�}_{k = 0}^{鈭�} (c r^{k} . b v^{k})$ converges. If neither c nor b is 0, could the series be $({鈭�}_{k = 0}^{鈭�} c r^{k}) ({鈭�}_{k = 0}^{鈭�} b v^{k})$ ?

Use either the divergence test or the integral test to determine whether the series in Given Exercises converge or diverge. Explain why the series meets the hypotheses of the test you select.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{1}{2} k^{鈭� 3 / 4}$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$2.2131313 . . .$

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