Q. 30 In Exercises 21-30聽use one of t... [FREE SOLUTION]

91影视

Chapter 7: Q. 30 (page 631)

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 1}^{鈭�} {(\frac{\sin k}{k})}^{2}$ .

Short Answer

Expert verified

The series ${鈭�}_{k = 1}^{鈭�} {(\frac{\sin k}{k})}^{2}$ is convergent.

Step by step solution

Step 1. Given information

${鈭�}_{k = 1}^{鈭�} {(\frac{\sin k}{k})}^{2}$ .

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = L$ , where $L$ is any positive real number, then either both converge or both diverge.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 0$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ also converges.
If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭�$ and ${鈭�}_{k = 1}^{鈭�} b_{k}$ diverges, then ${鈭�}_{k = 1}^{鈭�} a_{k}$ also diverges.

Step 3. The terms of the series ∑k=1∞ sin kk2 are positive.

The expression ${(\frac{\sin k}{k})}^{2}$ satisfies the following inequality,

$\frac{\sin^{2} k}{k^{2}} 鈮� \frac{1}{k^{2}}$ .

Find the series ${鈭�}_{k = 1}^{鈭�} b_{k}$ for the given series.

${鈭�}_{k = 1}^{鈭�} b_{k} = {鈭�}_{k = 1}^{鈭�} \frac{1}{k^{2}}$ .

The series ${鈭�}_{k = 1}^{鈭�} b_{k}$ is convergent by $p -$ series test.

Therefore, the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is also convergent.

Hence, the given series is also convergent.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 1}^{鈭�} {(\frac{\sin k}{k})}^{2}$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

Step 3. The terms of the series ∑k=1∞ sin kk2 are positive.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Decision Maths

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 21-30use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.鈭�k=1鈭�sinkk2.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

Step 3. The terms of the series ∑k=1∞ sin kk2 are positive.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Decision Maths

Statistics

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises $21 - 30$ use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.
${鈭�}_{k = 1}^{鈭�} {(\frac{\sin k}{k})}^{2}$ .