Section 11.1 Sequences Part I MAT 1236 Calculus III Section 11.1 Sequences Part I http://myhome.spu.edu/lauw

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

HW WebAssign 11.1 Part I

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.

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

Motivation

Motivation

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

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?

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.

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...

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?

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

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

Example One of the possible associated sequences of the series is

Example One of the possible associated sequences of the series is

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

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

Example 0(a)

Example 0(b)

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

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

Example 0(a)

Example 0(b)

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

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

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

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

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

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

Tool #1 (Theorem)

Tool #1 (Theorem) .

Tool #1 (Theorem) .

Tool #1 (Theorem)

Example 1 (a)

Example 1 (a)

Expectations

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

Example 1 (b)

Example 1 (b)

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

Standard Formula

Example 2

Expectations

Remark The following notation is NOT acceptable in this class

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

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

Tool #2 Use the Limit Laws and the formula

Example 3(a)

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

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

Example 3(b)

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