Chapter 7: Q. 65 (page 654)

Show that if a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ also converges.

Short Answer

Expert verified

To prove, we need to use the definition of absolute convergence.

Step by step solution

Step 1. Given information.

Consider the given question,

The series are ${鈭�}_{k = 1}^{鈭�} a_{k}, {鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ .

Step 2. Use the definition of absolute convergence.

Consider the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ which converges absolutely,

By the definition of convergence, $\lim_{k 鈫� 鈭�} |a_{k}| = 0$ .

Thus, there exists a positive integer Nsuch that $|a_{k}| < 1$ for $k > N$ .

For data-custom-editor="chemistry" $k > N$ ,

$0 鈮� |a_{k}^{2}| = |a_{k}| |a_{k}| |a_{k}^{2}| < |a_{k}|$

By comparison test, ${鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ converges as role="math" localid="1649265659139" ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

Therefore, if the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely then the series ${鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ converges.

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91影视

Show that if a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ also converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the definition of absolute convergence.

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91影视

Show that if a series 鈭�k=1鈭�akconverges absolutely, the series 鈭�k=1鈭�ak2 also converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the definition of absolute convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Show that if a series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}^{2}$ also converges.