Q. 13 Explain why every convergent ser... [FREE SOLUTION]

Chapter 7: Q. 13 (page 652)

Explain why every convergent series consisting of positive terms is absolutely convergent.

Short Answer

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The series consists of positive terms ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} |a_{k}|$ . It is absolutely convergent.

Step by step solution

Step 1. When the series consists of positive terms.

Consider the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is convergent consisting of positive terms.

Since, the series consists of positive terms, then ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} |a_{k}|$ .

Now according to the definition, the series is absolutely convergent, if the series ${鈭�}_{k = 1}^{鈭�} |a_{k}|$ converges.

Step 2. Check whether the series converges.

The given series to be absolutely convergent the series ${鈭�}_{k = 1}^{鈭�} |a_{k}|$ must converge.

As ${鈭�}_{k = 1}^{鈭�} a_{k} = {鈭�}_{k = 1}^{鈭�} |a_{k}|$ and series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is convergent, then ${鈭�}_{k = 1}^{鈭�} |a_{k}|$ is convergent.

By definition, the series ${鈭�}_{k = 1}^{鈭�} |a_{k}|$ is absolutely convergent.

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91影视

Explain why every convergent series consisting of positive terms is absolutely convergent.

Short Answer

Step by step solution

Step 1. When the series consists of positive terms.

Step 2. Check whether the series converges.

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