MAT 1236 Calculus III Section 11.1 Sequences Part I .
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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
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
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
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
If ( ) and lim ( ) then limn nx 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
If ( ) and lim ( ) then limn nx n
f n a f x L a L
Tool #1 (Theorem)
If ( ) and lim ( ) then limn nx n
f n a f x L a L
Example 1 (a)
nn
1lim
If ( ) and lim ( ) then limn nx n
f n a f x L a L
Example 1 (a)
nn
1lim
If ( ) and lim ( ) then limn nx n
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)
If ( ) and lim ( ) then limn nx n
f n a f x L a L
1
If 0, then lim 1n
np p
Example 1 (b)
If ( ) and lim ( ) then limn nx n
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
If ( ) and lim ( ) then limn nx n
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
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