Chapter 7: Q. 35 (page 657)

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.
$\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{1 + 2^{k}}$

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{1 + 2^{k}}$ diverges..

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{1 + 2^{k}}$

Step 2. Find limk→∞ak.

$\underset{k \to \infty}{l i m} a_{k} = \underset{k \to \infty}{l i m} (\frac{{(- 2)}^{k}}{1 + 2^{k}}) \neq 0$

Since the limit of the sequence does not converges to zero, so the sequence converge.

The sequence has to be either converge or diverge.

So the sequence diverges.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.
$\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{1 + 2^{k}}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find limk→∞ak.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Pure Maths

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.∑k=0∞(-2)k1+2k

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Find limk→∞ak.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Mechanics Maths

Pure Maths

Applied Mathematics

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.
$\sum_{k = 0}^{\infty} \frac{{(- 2)}^{k}}{1 + 2^{k}}$