Chapter 3: Sequences

Question 1

Prove the theorem that all convergent sequences are bounded

Answer 1

Question 2

What is the definition of a sequence ?

Answer 2

Question 3

Distinguish between the notation of a single element and an entire sequence

Answer 3

Question 4

What is the range of a sequence?

Answer 4

A set

Question 5

What is the definition of bounded?

Answer 5

Bounded above and bounded below

Question 6

What is the definition of convergence?

Answer 6

Question 7

What is the definition of divergence?

Answer 7

A sequence that does not converge

Question 8

Answer 8

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Answer 12

Question 13

Answer 13

Question 14

Prove the sum rule of convergence

Answer 14

Question 15

What is the sandwich Therom?

Answer 15

Question 16

What is the proof for the sandwich Therom

Answer 16

Question 17

Answer 17

Question 18

What is the proof for the product rule (2 attempts)?

Answer 18

Question 19

Answer 19

Question 20

Answer 20

Question 21

Answer 21

Question 22

What is a null sequence?

Answer 22

A sequence that converges to zero

Question 23

Answer 23

Question 24

Answer 24

Question 25

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Define what it means for a sequence to diverge to +- infinity

Answer 28

Question 29

Answer 29

Question 30

Define what it means for a sequence to be increasing, decreasing, strictly increasing and strictly decreasing

Answer 30

Question 31

What is a monotone sequence?

Answer 31

A sequence that is increasing or decreasing

Question 32

Proof the Therom

Answer 32

Question 33

Answer 33

Question 34

What is a corollary to the theorems? theorems: 1. If a sequence is increasing and bounded above then the sequence converges to the supreme. 2. if a sequence is decaresing and bounded below then it converges to the infinum.

Answer 34

theorems: 1. If a sequence is increasing and bounded above then the sequence converges to the supreme. 2. if a sequence is decaresing and bounded below then it converges to the infinum.

Question 35

Answer 35

Question 36

Define a subsequence

Answer 36

Question 37

Prove the theorem (three cases )

Answer 37

Question 38

Prove the lemma

Answer 38

Question 39

What is the Bolzano Weierstrass Theorem ?

Answer 39

Question 40

Prove the Bolzano Weierstrass theorem

Answer 40

Question 41

Answer 41

Question 42

What is the Cauchy schwarz inequality?

Answer 42

Question 43

Prove the Cauchy schwarz inequality

Answer 43

Question 44

What are the conditions for the following to be null sequences?

Answer 44

Question 45

Answer 45

Question 46

Answer 46

Question 47

Answer 47

Question 48

Answer 48

Question 49

Answer 49

Question 50

What is the proof for the quotient rule?

Answer 50

Question 51

What is the power rule? And proof

Answer 51

Question 52

Prove

Answer 52

Question 53

How does the nature of subsequences indicate convergence of main sequence?

Answer 53

Question 54

Question

Answer 54

Question 55

Question

Answer 55