Chapter 7: Q. 65 (page 654)

Show that if a series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely, the series $\sum_{k = 1}^{\infty} a_{k}^{2}$ 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 $\sum_{k = 1}^{\infty} a_{k}, \sum_{k = 1}^{\infty} a_{k}^{2}$ .

Step 2. Use the definition of absolute convergence.

Consider the series $\sum_{k = 1}^{\infty} a_{k}$ which converges absolutely,

By the definition of convergence, $\lim_{k \to \infty} |a_{k}| = 0$ .

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

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

$0 \leq |a_{k}^{2}| = |a_{k}| |a_{k}| |a_{k}^{2}| < |a_{k}|$

By comparison test, $\sum_{k = 1}^{\infty} a_{k}^{2}$ converges as role="math" localid="1649265659139" $\sum_{k = 1}^{\infty} a_{k}$ converges.

Therefore, if the series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely then the series $\sum_{k = 1}^{\infty} a_{k}^{2}$ converges.

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Show that if a series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely, the series $\sum_{k = 1}^{\infty} a_{k}^{2}$ also converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the definition of absolute convergence.

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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

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Show that if a series $\sum_{k = 1}^{\infty} a_{k}$ converges absolutely, the series $\sum_{k = 1}^{\infty} a_{k}^{2}$ also converges.