1. SEQUENCES DEF: A sequence is a list of numbers in a given order:
Example
first term
second term
n-th term
index
Example
Example

2. SEQUENCES Find the n-term Example Example Example Example Example
Even numbers
Example
odd numbers
Example
Example

3. SEQUENCES
Find the n-term
TERM-132

4. SEQUENCES
Find the n-term
TERM-092
TERM-131

5. Recursive Definitions
SEQUENCES
Recursive Definitions

6. Recursive Definitions
SEQUENCES
Example
Find a formula for the general term of the sequence
This sequence arose when the 13th-century Italian mathematician known as Fibonacci
Example
Find a formula for the general term of the sequence

7. SEQUENCES
Representation

8. SEQUENCES
Representing Sequences
Example

9. SEQUENCES
Limit of a Sequence

10. SEQUENCES Example Example LIMIT OF THE SEQUENCE as convg
Even numbers
What is the last number in the sequence ?
Example
What is the last number in the sequence ?
LIMIT OF THE SEQUENCE as
We say the sequence
convg

11. SEQUENCES LIMIT OF THE SEQUENCE as convg Remark: Remark:
We say the sequence
convg
If converges to L, we write
Remark:
If there exist no L then we say the sequence is divergent.
Remark:
or simply
and call L the limit of the sequence

12. 1 2 3 SEQUENCES Convergence or Divergence
How to find a limit of a sequence
Example
(IF you can)
use Math-101 to find the limit.
Use other prop.
To find the limit 1
Example: 2
1)Sandwich Thm:
2)Cont. Func. Thm: 3
3)L’Hôpital’s Rule:

13. SEQUENCES

14. SEQUENCES
TERM-132

15. SEQUENCES
Final-142
Final-141

16. SEQUENCES

17. SEQUENCES How to find a limit of a sequence
You find it in a sequence but not in a function
(IF you can)
use Math-101 to find the limit.
Use other prop.
To find the limit
Example:
1)Sandwich Thm:
2)Cont. Func. Thm:
3)L’Hôpital’s Rule:

18. SEQUENCES
Factorial;
NOTE
Example
REMARKS

19. SEQUENCES
Example
Find
where

20. SEQUENCES

21. SEQUENCES
Example
For what values of r is the sequence convergent?

22. SEQUENCES
Example:

23. Sec 11.1: SEQUENCES
Example
Note:

24. SEQUENCES How to find a limit of a sequence (convg or divg)
(IF you can)
use Math-101 to find the limit.
Use other prop.
To find the limit
abs,r^n,bdd+montone
Example:
Example:
1)Sandwich Thm:
1)Absolute value:
2)Cont. Func. Thm:
2)Power of r:
3)L’Hôpital’s Rule:
3)bdd+montone:
Bdd + monton  convg

25. SEQUENCES
Example
Find
Faster
Example
Find
where

27. SEQUENCES
Final-122

29. SEQUENCES

30. Multiple-Choice Problems
SEQUENCES
Extra
Multiple-Choice Problems

31. SEQUENCES

32. SEQUENCES

33. SEQUENCES

34. SEQUENCES

35. SEQUENCES

36. SEQUENCES
Final-141

37. SEQUENCES
TERM-092

38. SEQUENCES
TERM-092

39. SEQUENCES
TERM-082

40. SEQUENCES
If the sequence is convergent, guess the value of the limit and then prove your guess.

41. SEQUENCES Example: bounded unbounded unbounded bounded If is bounded
from above and below,
If is not bounded
we say that
bounded
unbounded
Example:
unbounded
bounded

42. SEQUENCES
Example
Find
where
Sol:
by sandw. limit is 0

43. Sec 11.1: SEQUENCES increasing decreasing DEFINITION Example Example
Is the sequence increasing or decreasing
Example
DEFINITION
decreasing
Is the sequence increasing or decreasing
2-solutions
Sol_1
Sol_2
DEFINITION
monotonic
if it is either increasing or decreasing.

44. Every bounded, monotonic sequence is convergent
Sec 11.1: SEQUENCES
DEFINITION
bounded above
Example
if there is a number M such that
Is the sequence bounded above or below
DEFINITION
bounded below
if there is a number m such that
DEFINITION
bounded
If it is bounded above and below
THEOREM;
Every bounded, monotonic sequence is convergent