Chapter 7: Q. 69 (page 654)

Prove that if the series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ diverges, then the series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ also diverges.

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ is convergent

The series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ is divergent

Step by step solution

Step 1. Given information

${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ and ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ are the given series

Step 2. Finding whether ∑k=1∞ akis convergent

Assume that ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ is not divergent,

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

If ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ is convergent but it is given that it is divergent.

If the series $\underset{k = 1}{鈭�} a_{k}$ is absolutely convergent, then the series $\underset{k = 1}{鈭�} a_{k}$ is convergent.

Therefore, ${鈭�}_{k = 1}^{鈭�} a_{k}$ is convergent, which is a contradiction as it is given that the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is divergent.

Therefore, the supposition that the series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ is not divergent is wrong.

Hence, the series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ is divergent.

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Prove that if the series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ diverges, then the series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ also diverges.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding whether ∑k=1∞ akis convergent

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91影视

Prove that if the series 鈭�k=1鈭�鈥�akdiverges, then the series 鈭�k=1鈭�鈥�akalso diverges.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding whether ∑k=1∞ akis convergent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Pure Maths

Statistics

Logic and Functions

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Prove that if the series ${鈭�}_{k = 1}^{鈭�} 鈥� a_{k}$ diverges, then the series ${鈭�}_{k = 1}^{鈭�} 鈥� |a_{k}|$ also diverges.