MAT 1236
Calculus III
Section 11.1
Sequences Part I
http://myhome.spu.edu/lauw
Continuous Vs Discrete
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An understand of discrete systems is
important for almost all modern
technology
HW
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WebAssign 11.1 Part I
(13 problems, 40* min.)
Quiz: 15.6-15.8, 11.1part I
Chapter 11
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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
3
5
7
x
x
x
sin x  x     
3! 5! 7!
2 n 1

x
n
  (1)
(2n  1)!
n 0
3
Motivation
x
sin x  x 
3!
3
Motivation
x
sin x  x 
3!
Foundations for Applications
Theory of
Series
Fourier
Series and
Transforms
Taylor
Series
Numerical
Analysis
Applications
in Sciences
and Eng.
Complex
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
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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...
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
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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

a
n 1
?
n
 a1  a2  a3    finite no. (convergen t)
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
a1 , a2 , a3 , 
Definition

A sequence is a collection of numbers
with an order
a1 , a2 , a3 , 

Notation:
an 
or
a 

n n 1
Example
One of the possible associated sequences of
the series
x 2 n1
sin x   (1)
(2n  1)!
n 0

n
is

x 2 n 1 
n
(1)

(2n  1)!

Example
One of the possible associated sequences of
the series
x 2 n1
sin x   (1)
(2n  1)!
n 0

n
is

x 2 n 1 
n
(1)

(2n  1)!

Another Example (Partial Sum
Sequence)
Another possible associated sequences of the
series
x 2 k 1
sin x   (1)
(2k  1)!
k 0

k
is
n
x 2 k 1 
k
 (1)

(2
k

1)!
 k 0

Another Example (Partial Sum
Sequence)
Another possible associated sequences of the
series
x 2 k 1
sin x   (1)
(2k  1)!
k 0

k
is
n
x 2 k 1 
k
 (1)

(2
k

1)!
 k 0

Example (Physics/Chemistry):
Balmer Sequence
364.5n 2
bn  2
, n3
n 4
The Balmer sequence plays a key role in
spectroscopy. The terms of the sequence
are the absorption wavelengths of the
hydrogen atom in nanometer.
Example 0(a)
1 
an    
n
 1 1 1 
 1, , , , 
 2 3 4 
an
Example 0(b)
 n  1
bn    
 n 
 3 4 5 
 2, , , , 
 2 3 4 
bn
Example 0
We want to know : As n  ,
an  ?
bn  ?
Use the limit notation, we want to know
lim an  ?
n
lim bn  ?
n 
Definition
A sequence {𝑎𝑛 } is convergent if
lim an  finite number
n 
Otherwise, {𝑎𝑛 }is divergent
Example 0(a)
1 
an    
n
 1 1 1 
 1, , , , 
 2 3 4 
lim an  0
n 
an
Example 0(b)
 n  1
bn    
 n 
 3 4 5 
 2, , , , 
 2 3 4 
lim bn  1
n 
bn
Example 0
We want to know : As n  ,
an  ?
bn  ?
In these cases,
lim an  0
n 
lim bn  1
n 
{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
lim (an  bn )  lim an  lim bn
n 
n 
n 
lim an  finite number
lim (anbn )  lim an  lim bn
n 
n 
n 
n 
lim (an / bn )  lim an / lim bn if lim bn  0
n 
n 
n 
n 
lim c  an  c  lim an
n 
n 
lim c  c
n 


lim an   lim an if p  0 and an  0
p
n 
n 
p
lim bn  finite number
n 
Remarks


lim an   lim an if p  0 and an  0
p
n 
p
n 
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

lim an   lim an
bn
n 
n 

lim bn
n
Remarks


lim an   lim an if p  0 and an  0
p
n 
n 
p
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
1
n
 2 
 2 
HW : lim 1     lim 1   
n 
 n   n  n  
1
n n
lim
Finding limits
There are 5 tools that you can use to find
limit of sequences
Tool #1 (Theorem)
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Tool #1 (Theorem)
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
f ( x), an
L
.
x, n
1 2
n
Tool #1 (Theorem)
If f (n)  an and lim f ( x)  L then lim an  L
x 
f ( x), an
n 
a2  f (2)
an  f (n)
L
.
a1  f (1)
x, n
1 2
n
Tool #1 (Theorem)
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Example 1 (a)
1
lim
n n
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Example 1 (a)
1
lim
n n
1
1
Let f ( x)  , then f (n) 
x
n
lim f ( x) 
x 
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Expectations
Standard Formula
In general, if 𝑟 > 0 is a rational number,
then
1
lim r  0
n n
Example 1 (b)
1
n
If p  0, then lim p  1
n 
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Example 1 (b)
1
n
If p  0, then lim p  1
n 
1
x
Let f ( x)  p , then f (n)  p
lim f ( x) 
x 
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
1
n
Remark: (2.5)
If lim g  x   b and the function 𝑓 is continuous
at 𝑏, then
x a


lim f  g  x    f lim g  x   f  b 
xa
x a
Standard Formula
1
n
If p  0, then lim p  1
n 
Example 2
n
lim n
n  e
If f (n)  an and lim f ( x)  L then lim an  L
x 
n 
Expectations
Remark
The following notation is NOT acceptable
in this class
x
lim f ( x)  lim x
x 
x  e
1
H lim x
x  e
PPFTNE Questions
Q: Can we use the l’ hospital rule on
sequences?
PPFTNE Questions
Q: Is the converse of the theorem also true?
If
f ( n)  a n
and
lim a n  L
n 
then
lim f ( x)  L
x 
If Yes, demonstrate why.
If No, give an example to illustrate.
Tool #2

Use the Limit Laws and the formula
1
lim r  0
n n
Example 3(a)
n 1
lim
n  2n 2  1
2
1
0
n n r
lim
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)
  n2  1  
lim sin   2
 

n 
  2n  1  
n2  1 1
lim 2

n  2n  1
2
Theorem
If lim a  L and the function 𝑓 is continuous
at 𝐿, then
n 
n
lim f  an   f  L 
n