1. Section 11.1 Sequences Part I
MAT 1236 Calculus III
Section 11.1
Sequences Part I

2. Continuous Vs Discrete
An understand of discrete systems is important for almost all modern technology

3. HW
WebAssign 11.1 Part I

4. Chapter 11 This chapter will be covered in the second and final exam.
Go over the note before you do your HW. Reading the book is very helpful.
If you are interested to become a math tutor, this is the chapter that you need to fully understand and master because...
DO NOT SKIP CLASSES.

5. Motivation Q: How to compute sin⁡(0.5)?
A: sin⁡(𝑥) can be computed by the formula

6. Motivation

7. Motivation

8. Foundations for Applications
Theory of Series
Taylor Series
Fourier Series and Transforms
Numerical Analysis
Complex Analysis
Applications in Sciences and Eng.

9. Caution
Most solutions of the problems in this chapter rely on precise arguments. Please pay extra attention to the exact arguments and presentations.
(Especially for those of you who are using your photographic RAM)-not in 2017?

10. Caution WebAssign HW is very much not sufficient in the sense that…
Unlike any previous calculus topics, you actually have to understand the concepts.
Most students need multiple exposure before grasping the ideas.
You may actually need to read the textbook, for the first time.

11. Come talk to me... I am not sure about the correct arguments...
I suspect the series converges, but I do not know why?
I think WebAssign is wrong...
I think my group is all wrong...
I have a question about faith...

12. General Goal We want to look at infinite sum of the form
Q: Can you name a concept in calculus II that involves convergent / divergent?

13. Sequences
Before we look at the convergence of the infinite sum (series), let us look at the individual terms

14. Definition A sequence is a collection of numbers with an order
Notation:

15. Example
One of the possible associated sequences of the series is

16. Example
One of the possible associated sequences of the series is

17. Another Example (Partial Sum Sequence)
Another possible associated sequences of the series is

18. Another Example (Partial Sum Sequence)
Another possible associated sequences of the series is

19. Example 0(a)

20. Example 0(b)

21. Example 0
We want to know : As
Use the limit notation, we want to know

22. Definition A sequence { 𝑎 𝑛 } is convergent if
Otherwise, { 𝑎 𝑛 }is divergent

23. Example 0(a)

24. Example 0(b)

25. Example 0 We want to know : As In these cases,
{𝑎𝑛}, {𝑏𝑛} are convergent sequences

26. Question Q: Name 2 divergent sequences
(with different divergent “characteristics”.)

27. Limit Laws
If {𝑎𝑛}, {𝑏𝑛} are 2 convergent sequences and 𝑐 is a constant, then

28. Remarks
Note that 𝑝 is a constant. If the power is not a constant, this law does not applied. For example, there is no such law as

29. Remarks
Note that 𝑝 is a constant. If the power is not a constant, this law does not applied. For example, there is no such law as

30. Finding limits
There are 5 tools that you can use to find limit of sequences

31. Tool #1 (Theorem)

32. Tool #1 (Theorem) .

33. Tool #1 (Theorem) .

34. Tool #1 (Theorem)

35. Example 1 (a)

36. Example 1 (a)

37. Expectations

38. Standard Formula
In general, if 𝑟>0 is a rational number, then

39. Example 1 (b)

40. Example 1 (b)

41. Remark:
If and the function 𝑓 is continuous at 𝑏, then

42. Standard Formula

43. Example 2

44. Expectations

45. Remark
The following notation is NOT acceptable in this class

46. PPFTNE Questions
Q: Can we use the l’ hospital rule on sequences?

47. PPFTNE Questions Q: Is the converse of the theorem also true?
If Yes, demonstrate why.
If No, give an example to illustrate. If
and
then

48. Tool #2
Use the Limit Laws and the formula

49. Example 3(a)

50. PPFTNE Questions
Q1: Can we use tool #1 to solve this problem?

51. PPFTNE Questions Q1: Can we use tool #1 to solve this problem?
Q2: Should we use tool #1 to solve this problem?

52. Example 3(b)

53. Theorem
If and the function 𝑓 is continuous at 𝐿, then