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Chapter 7: Q. 19 (page 652)

Give an example of divergent series ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ such that the series role="math" localid="1649434191353" ${鈭�}_{k = 1}^{鈭�} a_{k} b_{k}$ converges.

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} a_{k} b_{k} = {鈭�}_{k = 1}^{鈭�} (\frac{1}{k^{2}})$ is convergent.

Step by step solution

01

Step 1. Given information

${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are divergent.

02

Step 2. Check if ∑akk=1∞ converges

Calculating $\frac{a_{k + 1}}{a_{k}}$ ,

a_{k} = \frac{1}{k}

$a_{k + 1} = \frac{1}{(k + 1)}$

$\begin{aligned} \frac{a_{k + 1}}{a_{k}} = \frac{\frac{1}{(k + 1)}}{\frac{1}{k}} \\ = \frac{k}{(k + 1)} \end{aligned}$

Now, taking limits,

$\begin{aligned} lim_{k 鈫� 鈭�} 鈥� \frac{a_{k + 1}}{a_{k}} = lim_{k 鈫� 鈭�} 鈥� \frac{k}{(k + 1)} \\ = (lim_{k 鈫� 鈭�} 鈥� \frac{k}{(k + 1)}) \\ = 鈭� \end{aligned}$

Hence, the series role="math" localid="1649434828493" ${鈭� a_{k}}_{k = 1}^{鈭�} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k}$ diverges.

Similarly, ${鈭� b_{k}}_{k = 1}^{鈭�} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k}$ isdivergent.

03

Step 3. Check if ∑akbkk=1∞ is divergent or convergent

$\begin{aligned} {鈭�}_{k = 1}^{鈭�} 鈥� a_{k} b_{k} = {鈭�}_{k = 1}^{鈭�} 鈥� (\frac{1}{k}) (\frac{1}{k}) \\ = {鈭�}_{k = 1}^{鈭�} 鈥� (\frac{1}{k^{2}}) \end{aligned}$

Hence, ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k} b_{k}$ is convergent.

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Most popular questions from this chapter

Determine whether the series ${鈭�}_{k = 0}^{鈭�} 蟺 {(\frac{e}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?

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.

${鈭�}_{k = 1}^{鈭�} 鈥� k^{鈭� 3 / 2}$

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.

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.

${鈭�}_{k = 1}^{鈭�} \frac{1}{k}$

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