Chapter 7: Q. 18 (page 631)

If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$ and $\sum_{k = 1}^{\infty} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of $\sum_{k = 1}^{\infty} b_{k}$ .

Short Answer

Expert verified

$The behavior of the series {\sum b_{k}}_{k = 1}^{\infty} cannot be determined when \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and {\sum a_{k}}_{k = 1}^{\infty} is divergent holds because the series {\sum a_{k}}_{k = 1}^{\infty} may or may not be divergent, even though \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty holds .$

Step by step solution

Step 1. Given information

$\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty a n d \sum_{k = 1}^{\infty} a_{k} d i verges$

Step 2. Finding the value of limit

Consider the convergent series $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ and the series $\sum_{k = 1}^{\infty} b_{k} = \frac{1}{k^{2}}$ .

$The value of \lim_{k \to \infty} \frac{a_{k}}{b_{k}} is : \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \lim_{k \to \infty} \frac{\frac{1}{k}}{\frac{1}{k^{2}}} = \lim_{k \to \infty} k = \infty$

Step 3. Result

$The series {\sum b_{k}}_{k = 1}^{\infty} = \frac{1}{k^{2}} is convergent but the series {\sum a_{k}}_{k = 1}^{\infty} = \frac{1}{k} is divergent . The behavior of the series {\sum b_{k}}_{k = 1}^{\infty} cannot be determined when \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and {\sum a_{k}}_{k = 1}^{\infty} is divergent holds because the series {\sum a_{k}}_{k = 1}^{\infty} may or may not be divergent, even though \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty holds .$

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If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$ and $\sum_{k = 1}^{\infty} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of $\sum_{k = 1}^{\infty} b_{k}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of limit

Step 3. Result

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If limk→∞akbk=∞and∑k=1∞ak diverges, explain why we cannot draw any conclusions about the behavior of∑k=1∞bk.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Finding the value of limit

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

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If $\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$ and $\sum_{k = 1}^{\infty} a_{k}$ diverges, explain why we cannot draw any conclusions about the behavior of $\sum_{k = 1}^{\infty} b_{k}$ .