Q. 6 Explain how you could adapt the ... [FREE SOLUTION]

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Chapter 7: Q. 6 (page 631)

Explain how you could adapt the limit comparison test to analyze a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which all of the terms are negative.

Short Answer

Expert verified

To adapt limit comparison, apply it on ${鈭�}_{k = 1}^{鈭�} (- a_{k})$ because $- a_{k}$ is positive for allk

Step by step solution

Step 1. Given information

A series is given as ${鈭�}_{k = 1}^{鈭�} a_{k}$

Step 2. Limit comparison test

The limit comparison test for ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are the series having positive terms the the following conditions may apply,

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , L must be positive number then it may be either converging or diverging.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ then if ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ then if ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges then ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges

Now the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ has all terms negative, therefore $- a_{k}$ is positive for all k.

To adapt limit comparison, apply it on ${鈭� (-}_{k = 1}^{鈭�} a_{k})$ because $- a_{k}$ is positive.

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91影视

Explain how you could adapt the limit comparison test to analyze a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which all of the terms are negative.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit comparison test

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91影视

Explain how you could adapt the limit comparison test to analyze a series 鈭�k=1鈭�ak in which all of the terms are negative.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit comparison test

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Statistics

Pure Maths

Logic and Functions

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Explain how you could adapt the limit comparison test to analyze a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which all of the terms are negative.