Chapter 7: Q. 16 (page 631)

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

Short Answer

Expert verified

$The behaviour of the series {鈭�}_{k = 1}^{鈭�} b_{k} cannot be determined when \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0 and {鈭�}_{k = 1}^{鈭�} a_{k} is convergent holds because the series {鈭�}_{k = 1}^{鈭�} b_{k} may or may be not convergent, even though \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0 holds .$

Step by step solution

Step 1. Given information

$\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k} Converges$

Step 2. Explanation

${鈭�}_{k = 1}^{鈭�} a_{k} = \frac{1}{k^{2}} and the series {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k} The value of \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = \lim_{k 鈫� 鈭�} \frac{\frac{1}{k^{2}}}{\frac{1}{k}} \lim_{k 鈫� 鈭�} \frac{1}{k} = 0$

Step 3. Result

${鈭�}_{k = 1}^{鈭�} a_{k} = \frac{1}{k^{2}} i s divergen t and the series {鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k} is convergent The behaviour of the series {鈭�}_{k = 1}^{鈭�} b_{k} cannot be determined when \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0 and {鈭�}_{k = 1}^{鈭�} a_{k} is convergent holds because the series {鈭�}_{k = 1}^{鈭�} b_{k} may or may be not convergent, even though \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0 holds .$

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If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Result

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Iflimk鈫�鈭�akbk=鈭�and鈭�k=1鈭�akconverges, explain why we cannot draw any conclusions about the behavior of鈭�k=1鈭�bk.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

Step 3. Result

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Most popular questions from this chapter

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If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� a n d {鈭�}_{k = 1}^{鈭�} a_{k}$ converges, explain why we cannot draw any conclusions about the behavior of ${鈭�}_{k = 1}^{鈭�} b_{k}$ .