Chapter 7: Q. 2 (page 624)

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .
(b) A divergent p-series.
(c) A convergent p-series.

Short Answer

Expert verified

(a) The example of the series is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ .

(b) The example of the series is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ .

Step by step solution

Part (a) Step 1. Given Information.

A divergent series:

$\sum_{k = 1}^{\infty} a_{k}$

And $a_{k} \to 0$

Part (a) Step 2. Consider the given series.

Consider the given series.

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k} = \underset{k \to \infty}{l i m} \frac{1}{k} = 0$

Part (a) Step 3. Find the series.

So by using the harmonic series and p-test series, the series $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$ is divergent.

Part (b) Step 1. Find an example.

Consider the series,

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k}$

which is a harmonic series, and by p-series test, the series is divergent.

Part (c) Step 1. Consider the series.

Consider the series,

$\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$

which is convergent since $p = 2 > 1$ .

So the convergent p-series test is $\sum_{k = 1}^{\infty} a_{k} = \sum_{k = 1}^{\infty} \frac{1}{k^{2}}$ .

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.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .
(b) A divergent p-series.
(c) A convergent p-series.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Consider the given series.

Part (a) Step 3. Find the series.

Part (b) Step 1. Find an example.

Part (c) Step 1. Consider the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.(a) A divergent series ∑k=1∞akin which ak→0.(b) A divergent p-series.(c) A convergent p-series.

Short Answer

Step by step solution

Part (a) Step 1. Given Information.

Part (a) Step 2. Consider the given series.

Part (a) Step 3. Find the series.

Part (b) Step 1. Find an example.

Part (c) Step 1. Consider the series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .
(b) A divergent p-series.
(c) A convergent p-series.