Sequences

1. Sequences and limits.

Definition 1.1 A sequence of real numbers is a function .

Definition 1.2 The sequence is called increasing (respectively decreasing) if (respectively ). Increasing sequences and decreasing sequences are called monotonic sequences.

Definition 1.3 The sequence is bounded if and only if such that .

Notation: ;

Definition 1.5 The sequence is convergent, , if such that .

Definition 1.6 if such that .

Definition 1.7 if such that

If , then the sequence is divergent.

A sequence can be defined in several ways:

• descriptively (e.g. 1,3,5,7,9,…);

• by a formula for the general term (e.g. );

• by a recurrence formula (e.g. , , ).

2. Sufficient criteria for convergence, or for a sequence to have a limit.

1. if and then ;

2. if and , then

3. if and , then ;

4. every monotonic bounded sequence is convergent (Weierstrass's criterion);

5. if and then ;

6. Stolz's criterion:

- if is increasing, and exists, then

- if , and exists, then (Cesàro).

- if is decreasing, and exists, then .

7. The ratio test.

Let be a sequence of strictly positive numbers. Suppose exists:

• if ,

• if .

3. Operations with convergent sequences

1.;

2.;

3. (if ).

4. Operations with sequences that have a limit.

1. if and then , ;

2. if then ;

3. if and then ,

4. if then ;

5. if and , then ;

6. if then if and if .

6. Indeterminate forms

6. Standard sequences

1.

2.

3. ;

4.;

5.;

6.;

7.;

8.;

9.;

10. ;

11. ;

12. ;

13. , ;

14. ;

15. .

Limits of functions

Notation: – an accumulation point of D ;

1. Definitions of the limit

Definition 1.1 , if for every neighbourhood V of l there is a neighbourhood U of such that implies ;

Definition 1.2 , if for every sequence , , with it follows that (the sequential criterion);

Definition 1.3 , if such that and imply ;

Definition 1.4 , if , where and .

2. Operations with limits of functions

– an accumulation point of D, ;

1.;

2.;

3.

4. if

3. Standard limits

1.;

;

2.

;

3.;

;

4.;

;

5. finite,

and if ;

and if

6., ;

;

;

;

7.;

;

;

;

;

;

;

8.

9.;

10.;

11.;

12.

Graph of the function

for r=3

13..