Q. 15 What does it mean for a sequence... [FREE SOLUTION]

Chapter 7: Q. 15 (page 591)

What does it mean for a sequence $\{a_{k}\}$ to be bounded above? Bounded below? Bounded?

Short Answer

Expert verified

If the sequence $\{a_{k}\}$ and a real number $M$ such that $M > a_{k}$ for every $k$ , then the sequence is bounded above.

If the sequence $\{a_{k}\}$ and a real number $M$ such that $M < a_{k}$ for every $k$ , then the sequence is bounded below.

The sequence which is both bounded above and bounded below is called a bounded sequence.

Step by step solution

Step 1. Given information

The given sequence is, $\{a_{k}\}$ .

Step 2. Consider the sequence ak,

And a real number $M$ such that $M > a_{k}$ for every $k$ , then the sequence is bounded above.

Consider the sequence ${\{- n^{2}\}}_{n = 0}^{鈭�}$

The terms of the sequence are, $\{0, - 1, - 4, . . . .\}$ .

The sequence is bounded above by $0$ .

Hence, the sequence ${\{- n^{2}\}}_{n = 0}^{鈭�}$ is bounded above.

Step 3. Consider the sequence ak,

And a real number $M$ such that $M < a_{k}$ for every $k$ , then the sequence is bounded below.

Consider the sequence ${\{2^{n}\}}_{n = 0}^{鈭�}$

The terms of the sequence are, $\{1, 4, 8, . . .\}$ .

The sequence is bounded above by $1 .$

Hence, the sequence ${\{2^{n}\}}_{n = 0}^{鈭�}$ is bounded below.

Step 4. The sequence which is both bounded above and bounded below is called a bounded sequence.

Consider the sequence ${\{{(- 1)}^{n + 1}\}}_{n = 1}^{鈭�}$

The terms of the sequence are, $\{- 1, 1, - 1, 1, . . . .\}$ .

The sequence is bounded below by $- 1$ and bounded above by $1 .$

Hence, the sequence ${\{{(- 1)}^{n + 1}\}}_{n = 1}^{鈭�}$ is a bounded sequence.

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

What does it mean for a sequence $\{a_{k}\}$ to be bounded above? Bounded below? Bounded?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider the sequence ak,

Step 3. Consider the sequence ak,

Step 4. The sequence which is both bounded above and bounded below is called a bounded sequence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

What does it mean for a sequence akto be bounded above? Bounded below? Bounded?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider the sequence ak,

Step 3. Consider the sequence ak,

Step 4. The sequence which is both bounded above and bounded below is called a bounded sequence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

What does it mean for a sequence $\{a_{k}\}$ to be bounded above? Bounded below? Bounded?