mat 1236 calculus iii section 11.1 sequences part i

MAT 1236Calculus 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(13 problems, 40* min.)

Quiz: 15.6-15.8, 11.1part 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.

For those of you who want to become a math tutor, this is the chapter that you need to fully understand.

DO NOT SKIP CLASSES.

Motivation

Q: How to compute sin(0.5)?

A: sin(x) can be computed by the formula

0

12

753

)!12()1(

!7!5!3sin

n

nn

n

x

xxxxx

Motivation !3sin

3xxx

Foundations for Applications

Theory of Series

Applications in Sciences

and Eng.

Taylor Series

Fourier Series and Transforms

Complex Analysis

Numerical Analysis

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)

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...

Chocolate in my office

General Goal

We want to look at infinite sum of the form

t)(convergen no. finite ?

3211

aaaan

n

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

, , , 321 aaa

Definition

A sequence is a collection of numbers with an order

, , , 321 aaa

Notation:

na 1nnaor

Example

One of the possible associated sequences of the series

is

2 1

0

sin ( 1)(2 1)!

nn

n

xx

n

2 1

( 1)(2 1)!

nn x

n

Another Example (Partial Sum Sequence)

Another possible associated sequences of the series

is

2 1

0

sin ( 1)(2 1)!

kk

k

xx

k

2 1

0

( 1)(2 1)!

knk

k

x

k

Example (Physics/Chemistry):Balmer Sequence

The Balmer sequence plays a key role in spectroscopy. The terms of the sequence are the absorption wavelengths of the hydrogen atom in nanometer.

2

2

364.5, 3

4n

nb n

n

Example 0(a)

,4

1,

3

1,

2

1,1

1

nan

na

Example 0(b)

,4

5,

3

4,

2

3,2

1

n

nbn

nb

Example 0

We want to know : As ,n

?na ?nb

?lim n

na ?lim

nnb

Use the limit notation, we want to know

Definition

A sequence is convergent if

number finitelim n

na

Otherwise,is divergent

Example 0(a)

,4

1,

3

1,

2

1,1

1

nan

na

0lim n

na

Example 0(b)

,4

5,

3

4,

2

3,2

1

n

nbn

nb

1lim n

nb

Example 0

We want to know : As ,n

?na ?nb

In these cases,

0lim n

na 1lim

nnb

, are convergent sequences

Question

Q: Name 2 divergent sequences

(with different divergent “characteristics”.)

0 and 0 if limlim

lim

limlim

0lim if lim/lim)/(lim

limlim)(lim

limlim)(lim

np

nn

pn

n

n

nn

nn

nn

nn

nn

nnn

nn

nn

nnn

nn

nn

nnn

apaa

cc

acac

bbaba

baba

baba

Limit LawsIf , are 2 convergent sequences and is a constant, then

lim finite numbernna

lim finite numbernnb

Remarks

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

0 and 0 if limlim n

pn

n

pn

napaa

limlim lim nn

n bn

n

bn

naa

Remarks

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

0 and 0 if limlim n

pn

n

pn

napaa

11 lim2 2

: lim 1 lim 1 n nn

n nHW

n n

Finding limits

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

Tool #1 (Theorem)

If ( ) and lim ( ) then limn nx n

f n a f x L a L

Tool #1 (Theorem)

.

naxf ),(

nx,

L

1 2 n


f n a f x L a L

Tool #1 (Theorem)

.

naxf ),(

nx,

L

1

)1(1 fa

)2(2 fa

)(nfan

2 n


f n a f x L a L

Tool #1 (Theorem)


f n a f x L a L

Example 1 (a)

nn

1lim


f n a f x L a L

Example 1 (a)

nn

1lim


f n a f x L a L

1 1Let ( ) , then ( )

lim ( )x

f x f nx n

f x

Expectations

Standard Formula

01

lim rn n

In general, if is a rational number, then

Example 1 (b)


f n a f x L a L

1

If 0, then lim 1n

np p

Example 1 (b)


f n a f x L a L

1

If 0, then lim 1n

np p

1 1

Let ( ) , then ( )

lim ( )

x n

x

f x p f n p

f x

Remark: (2.5)

If and the function is continuous at , then

lim limx a x a

f g x f g x f b

limx ag x b

Standard Formula

1

If 0, then lim 1n

np p

Example 2

nn e

n

lim


f n a f x L a L

Expectations

Remark

The following notation is NOT acceptable in this class

lim ( ) lim

1lim

xx x

xx

xf x

e

He

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 nanf )( Lxfx

)(lim then Lan

n

lim and

Tool #2

Use the Limit Laws and the formula

01

lim rn n

Example 3(a)

12

1lim

2

2

n

nn

01

lim rn n

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)

2

2

1lim sin

2 1n

n

n

2

2

1 1lim

2 1 2n

n

n

Theorem

If and the function is continuous at , then

lim nn

f a f L

Lann

lim

mat 1236 calculus iii section 11.1 sequences part i

Documents

b example

theorem example

example physicschemistry

definition example

example partial sum

possible associated

divergent sequences

infinite sum series