Chapter 7: Q. 17 (page 631)

Consider the series
Fill in the blanks and select the correct word:
$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____diverges t he n \sum_{k = 1}^{\infty}_____(converges / diverges) .$

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

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

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Consider the series
Fill in the blanks and select the correct word:
$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____diverges t he n \sum_{k = 1}^{\infty}_____(converges / diverges) .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Limit comparison test

Step 3. Result

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Consider the seriesFill in the blanks and select the correct word:Iflimk→∞akbk=∞and∑k=1∞_____divergesthen∑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

Calculus

Statistics

Geometry

Theoretical and Mathematical Physics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Consider the series
Fill in the blanks and select the correct word:
$I f \lim_{k \to \infty} \frac{a_{k}}{b_{k}} = \infty and \sum_{k = 1}^{\infty}_____diverges t he n \sum_{k = 1}^{\infty}_____(converges / diverges) .$