Q. 17 Let聽ak聽be a sequence of positi... [FREE SOLUTION]

Chapter 7: Q. 17 (page 652)

Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}$ and ${鈭�}_{k = 1}^{鈭�} a_{2 k}$ both converge absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely.

Short Answer

Expert verified

As the given series is absolutely convergent and ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}$ is the series of odd numbers and ${鈭�}_{k = 1}^{鈭�} a_{2 k}$ is the series of even numbers.

Step by step solution

. Given information.

Consider the given question,

$\{a_{k}\}$ is the sequence of positive numbers.

Step 2. Explain the series.

The series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}$ is the series of odd numbers ${鈭�}_{k = 1}^{鈭�} a_{2 k}$ is the series of even numbers. Both the series converge absolutely.

The series ${鈭�}_{k = 1}^{鈭�} a_{k}$ is the sum of series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}, {鈭�}_{k = 1}^{鈭�} a_{2 k}$ .

As the sum of two convergent series is convergent.

Moreover, the series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}, {鈭�}_{k = 1}^{鈭�} a_{2 k}$ is absolutely convergent.

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

Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}$ and ${鈭�}_{k = 1}^{鈭�} a_{2 k}$ both converge absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely.

Short Answer

Step by step solution

. Given information.

Step 2. Explain the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Geometry

Discrete Mathematics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Letakbe a sequence of positive numbers. Explain why, if the series鈭�k=1鈭�a2k-1and鈭�k=1鈭�a2kboth converge absolutely, the series鈭�k=1鈭�akconverges absolutely.

Short Answer

Step by step solution

. Given information.

Step 2. Explain the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Pure Maths

Geometry

Discrete Mathematics

Statistics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Let $\{a_{k}\}$ be a sequence of positive numbers. Explain why, if the series ${鈭�}_{k = 1}^{鈭�} a_{2 k - 1}$ and ${鈭�}_{k = 1}^{鈭�} a_{2 k}$ both converge absolutely, the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges absolutely.