1. Convergent and divergent sequences. Limit of sequence.
Sandwich theorem.
Rita Korsunsky

2. Sequences Example 1 Sequence Notation: {an} =a1, a2,a3,…,an,…
A sequence is a function f whose domain is the set of positive integers
Sequence Notation:
{an} =a1, a2,a3,…,an,…
For example: {2n}= 21, 22, 23,…,2n,…
Example 1
List the first four terms and the tenth term of each sequence:
Sequence an a1, a2, a3, a a10
a. {2 + (0.1)n}
2 + (0.1)n , 2.01, 2.001,
c. {4}
, 4, 4,

3. Recursive Definition of the Sequence
A sequence is defined recursively when each term following the first term in the sequence is defined with the previous term in the sequence
Example 2: Find the first four terms and the nth term of the sequence : a1 = 3 and ak+1 = 2ak for k>1

4. Theorem Example 3 Sequence {an} converges to 1
Let {an} be a sequence, f(n) = an, and f(x) exists for every real x>1
Example 3
Sequence {an} converges to 1

5. Example 4 The sequence converges to 0
Determine whether the sequence converges or diverges
The sequence diverges
(b) {(-1)n-1}
1,-1,1,-1,1,-1,1,…
The sequence diverges
Using L’Hopital’s Rule
The sequence converges to 0

6. Theorem Example 5 Example 6 r = 1.01>1
List the first three terms of the sequence, and determine if it converges or diverges:
sequence converges to 0
(b) {1.01n}
r = 1.01>1
1.01, ,
sequence diverges
Example 6
Find the limit of the sequence

7. The Sandwich Theorem then Example 8:
If {an}, {bn}, and {cn}are sequences and an<bn<cn for every n and if
then
Example 8:
Find the limit of the sequence {cos2n/3n}
0cos2n 1 for every positive integer n, so
The limit of the sequence is 0

8. Theorem Example 9 Let {an} be a sequence.
Proof:
Example 9
Suppose the nth term of a sequence is
1, -1/2, 1/3, -1/4, 1/5, -1/6, 1/7

9. Definitions Theorem A bounded, monotonic sequence has a limit
A sequence is monotonic if successive terms are decreasing or increasing or all equal
A sequence is bounded if there is a positive real number M such < M for every k
Theorem
A bounded, monotonic sequence has a limit