Q. 18 If limk鈫掆垶akbk聽=聽鈭灺燼nd聽... [FREE SOLUTION]

Chapter 7: Q. 18 (page 631)

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

Short Answer

Expert verified

$The behavior of the series {鈭� b_{k}}_{k = 1}^{鈭�} cannot be determined when \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� and {鈭� a_{k}}_{k = 1}^{鈭�} is divergent holds because the series {鈭� a_{k}}_{k = 1}^{鈭�} may or may not be divergent, even though \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� holds .$

Step by step solution

Step 1. Given information

$\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k} d i verges$

Step 2. Finding the value of limit

Consider the convergent series ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k}$ and the series ${鈭�}_{k = 1}^{鈭�} b_{k} = \frac{1}{k^{2}}$ .

$The value of \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} is : \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{1}{k}}{\frac{1}{k^{2}}} = \lim_{k 鈫� 鈭�} k = 鈭�$

Step 3. Result

$The series {鈭� b_{k}}_{k = 1}^{鈭�} = \frac{1}{k^{2}} is convergent but the series {鈭� a_{k}}_{k = 1}^{鈭�} = \frac{1}{k} is divergent . The behavior of the series {鈭� b_{k}}_{k = 1}^{鈭�} cannot be determined when \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� and {鈭� a_{k}}_{k = 1}^{鈭�} is divergent holds because the series {鈭� a_{k}}_{k = 1}^{鈭�} may or may not be divergent, even though \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� holds .$

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If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of limit

Step 3. Result

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91影视

If limk鈫�鈭�akbk=鈭�and鈭�k=1鈭�ak diverges, explain why we cannot draw any conclusions about the behavior of鈭�k=1鈭�bk.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of limit

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Decision Maths

Statistics

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Calculus

Study anywhere. Anytime. Across all devices.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .