Chapter 7: Q. 15 (page 631)

$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____Converges the n \sum_{k = 1}^{\infty}_____(converges / diverges) .$

Short Answer

Expert verified

$The first blank is filled with \sum_{k = 1}^{\infty} b_{k} and second with \sum_{k = 1}^{\infty} a_{k} .$

Step by step solution

Step 1. Given information

$\lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty$

Step 2.. Limit comparison test

$The limit comparison test states that for and be two series with positive terms then If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = L Where L is any positive real number then either both converges or diverges . If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = 0 and \sum_{k = 1}^{\infty} b_{k} converges then \sum_{k = 1}^{\infty} a_{k} converges . If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty} b_{k} diverges then \sum_{k = 1}^{\infty} a_{k} diverges .$

Step 3. Result

$If \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty} b_{k} Converges then \sum_{k = 1}^{\infty} a_{k} Converges Therefore, the first blank is filled with \sum_{k = 1}^{\infty} b_{k} and second with \sum_{k = 1}^{\infty} a_{k} .$

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$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____Converges the n \sum_{k = 1}^{\infty}_____(converges / diverges) .$

Short Answer

Step by step solution

Step 1. Given information

Step 2.. Limit comparison test

Step 3. Result

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Iflimk→∞akbk=∞and∑k=1∞_____Convergesthen∑k=1∞_____(converges/diverges).

Short Answer

Step by step solution

Step 1. Given information

Step 2.. Limit comparison test

Step 3. Result

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Probability and Statistics

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Study anywhere. Anytime. Across all devices.

$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____Converges the n \sum_{k = 1}^{\infty}_____(converges / diverges) .$