Chapter 7: Q. 55 (page 632)

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are two convergent series with $a_{k} 鈮� 0$ and $b_{k} 鈮� 0$ for every positive integer k, then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$ converges.

Short Answer

Expert verified

${鈭�}_{k = 1}^{鈭�} b_{k}$ is convergent, so according to the divergence test, $\lim_{k 鈫� 鈭�} b_{k} = 0 .$

Then

$\lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = \lim_{k 鈫� 鈭�} b_{k} \lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = 0$

According to The Limit Comparison Test, if $\lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$ also converges.

Step by step solution

Step 1. Given Information.

Series ${鈭�}_{k = 1}^{鈭�} a_{k} & {鈭�}_{k = 1}^{鈭�} b_{k}$ is a convergent where $a_{k} 鈮� 0 & b_{k} 鈮� 0$ for every positive integer k.

The given series are the following.

localid="1649095185448" ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$

Step 2. Proof

Consider $\lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = \lim_{k 鈫� 鈭�} b_{k} .$

${鈭�}_{k = 1}^{鈭�} b_{k}$ is convergent, so according to the divergence test, $\lim_{k 鈫� 鈭�} b_{k} = 0 .$

So role="math" localid="1649095132998" $\lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = \lim_{k 鈫� 鈭�} b_{k} \lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = 0$

According to The Limit Comparison Test, if $\lim_{k 鈫� 鈭�} \frac{a_{k} 路 b_{k}}{a_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$ converges.

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91影视

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are two convergent series with $a_{k} 鈮� 0$ and $b_{k} 鈮� 0$ for every positive integer k, then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$ converges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof

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91影视

Prove that if 鈭�k=1鈭�akand 鈭�k=1鈭�bkare two convergent series with ak鈮�0and bk鈮�0for every positive integer k, then the series 鈭�k=1鈭�ak路bkconverges.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Proof

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ are two convergent series with $a_{k} 鈮� 0$ and $b_{k} 鈮� 0$ for every positive integer k, then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} 路 b_{k})$ converges.