CHAPTER 1

SEQUENCES AND INFINITE SERIES

SEQUENCES AND INFINITE SERIES (10 meetings)

Sequences and limit of a sequence

Monotonic and bounded sequence

Infinite series of constant terms

Infinite series of positive terms

Alternating series

Power series

Differentiation and Integration of Power Series

Taylor and Maclaurin series

CHAPTER OBJECTIVE

At the end of the chapter, you should be able

to:

1.Determine if a given sequence is convergent or

divergent.

2.Determine if a given series is convergent or

divergent.

3.Differentiate/integrate an infinite series.

CHAPTER OBJECTIVE

At the end of the chapter, you should be able

to:

4. Find the interval and radius of convergence of

a given series.

5. Write the Maclaurin/Taylor series expansion

of a function.

1.1 Sequences

A sequence of real numbers

is a function that assigns to each positive integer

a number .

DOMAIN:

The numbers in the range are called the elements

or terms of the sequence.

1 2, ,..., ,...na a a

n na

N Some books use Domain: W

1.1 Sequences

NOTATIONS:

1n n

a

na

f n

Whats next in the sequence?

0, 3, 8, 15, 24, 35,

1, 1, 2, 3, 5, 8, 13,

1 1 3 1 3 5 1 3 5 7, , , ,

2 2 4 2 4 6 2 4 6 8

1 3 5 7 9

2 4 6 8 10

48

21

NOTE Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, Iterative/Recursive Relation (difference equation):

+ 2 = + 1 + , (1) = 1, (2) = 1

General term (solution to the difference equation):

=5 + 5

10

1 + 5

2

1

+5 5

10

1 5

2

1

FYI: lim

(+1)

()= =

1+ 5

2= 1.618 (golden ratio)

Recurrence formula

Explicit formula

OUR INTEREST IN SEQUENCES:

Behavior of f nnas

Let . limn

f n L

OUR INTEREST IN SEQUENCES:

Some indicators of existence of limit:

increasing or decreasing

bounded

monotonicity is not necessary

boundedness is necessary but not sufficient

Example 1.

Let . 2 1f n n

n

f n

1 2 3 4 5 6 7

0 3 8 15 24 35 48

n f n

1 2 3 4 5 6 7

0 3 8 15 24 35 48

110

Example 2.

Let . 1

1n

g n

n

g n

1 2 3 4 5 6 7

1 1 1 1 1 1 1

n g n

1 2 3 4 5 6 7

1

1 1 1 1 1 1 1

1

1

Example 3.

Let . nh n e

n

h n

1 2 3 4 5 6 7

1

e 21

e 31

e4

1

e 51

e 61

e 71

e

n h n

1 2 3 4 5 6 7

1

e 21

e 31

e4

1

e 51

e 61

e 71

e

1

Example 4.

Let . 1

1n

j nn

n

j n

1 2 3 4 5 6 7

11

2

1

3

1

4

1

5

1

6

1

7

n j n

1 2 3 4 5 6 7

11

2

1

3

1

4

1

5

1

6

1

7

1

1

1

1

1

where 3.9 1

assume [0,1]

n n n nx x x x

x

Example 5. Try this in MS Excel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50

Chaos!

The Limit of a Sequence

The limit of a sequence f is the real

number L if for any , however small,

there exists a number such that if

is a natural number and if ,

then .

0

0N

Nn

Lnf

n

We write: limn

f n L

Example Consider lim

1

1

= 1

For any real number > 0, take

=1

If > , then

>1

>1

> 1

> 1 1

1 .

We need to find this Illustration: Suppose = 0.01

=1

0.01= 100

Hence, for all > 100, the

distance 1 1

1 is less

than 0.01.

Theorem.

If and is defined for every

positive integer then .

limx

f x L

f

limn

f n L

Recall: lim 0n

ne

Note that is defined for

every positive integer and .

xf x e

lim 0x

xe

Definition.

If in , exists,

Then the sequence is said to be convergent.

Otherwise it is divergent.

limn

f n L

L

Which of the ff sequences is/are

convergent?

3 4

n

n

2

tanArc nn

71

n

n

3 4cos

2

n

n n

11 n !10

n

3

2 !

n

n

NOTE Speed of functions, ranking: - constant (e.g. 10) - logarithmic (e.g. log n, log(n2)) - fractional power (e.g. sqrt(n)) - linear (e.g. n, 5n+10) - loglinear (e.g. n log n, log n!) - quadratic (e.g. n2, 7n2+9) - cubic (e.g. n3, 8n3+5n+2) - higher degree polynomials (FYI: 2log n is as fast as polynomials) - exponential (e.g., 2n, 1.1n2, nn) where base>1 - factorial (e.g. n!, 2n!+3)

Can you still remember how to get horizontal asymptote?

Use LHR!

1.2 Monotonic and Bounded Sequences

Monotone Convergence Theorem (MCT)

for Sequences.

A bounded monotonic sequence is

convergent.

When are sequences monotonic?

bounded?

1.2 Monotonic and Bounded Sequences

Definitions.

A sequence is monotonic if it is either

increasing or decreasing for all n.

A sequence is monotone increasing if na1 ,n na a n N

A sequence is monotone decreasing if na1 ,n na a n N

How do we determine if a sequence is

monotonic or not?

1

n

n

a

a

1. Observe .

2. Obtain . Then Compare result to

1(one).

2. Find .

'f x

na

Definitions.

A sequence is bounded if it has both

an upper bound and a lower bound.

A real number is a lower bound

of the sequence if

l

,nl a n N

A lower bound is the greatest lower

bound (glb) of the sequence if for all

lower bound .

l g

g

l

Definitions.

A real number is an upper bound

of the sequence if

u

,nu a n N

An upper bound is the least upper bound

(lub) of the sequence if for all upper

bound .

u v

v

u

Example 1. 5 1

2

n

n

5 1

2

xf x

x

Let

Since ,

2

2'

4f x

x

' 0 1f x x f is decreasing.

Now, . 5 1

02

n

n

f has 0 as a lower bound (5/2

is the glb)

and 3 as an upper bound.

Thus, the sequence is monotonic and bounded.

Example 2. !

10

n

!

10n

na Let

1

1 !

10n

na

Now, 1

! 10

10 1 !

n

n

a n

a n

1

1n

1

That is, 1 1n na a n

Thus, the sequence is monotonic (increasing).

Example 2. !

10

n

Thus, the sequence is unbounded.

Note that . !

010

n

has 0 as a lower bound (1/10 is the glb)

but has no upper bound.

!

10

n

Example 3. 11 n

n

na

1 2 3 4 5 6 7

1 1 1 1 1 1 1

Recall:

Thus, the sequence is bounded but is neither

increasing nor decreasing.

Example 4.

3

2 !

n

n

3

2 !

n

nan

Let

1

13

3 !

n

nan

Now,

1

1

3 !3

2 ! 3

nn

nn

a n

a n

33

n

That is, 1 1n na a n

Thus, the sequence is monotonic (decreasing).

1

Example 4.

3

2 !

n

n

Thus, the sequence is bounded.

Note that .

30

2 !

n

n

has 0 as a lower bound

and has as an upper bound. 3

2 !

n

n

REMARKS:

A bounded monotone decreasing sequence

converges to its greatest lower bound.

Similarly, a bounded monotone increasing

sequence converges to its least upper

bound.

Example (MCT is not applicable but has a

limit):

Let . 1

1n

j nn

n

j n

1 2 3 4 5 6 7

11

2

1

3

1

4

1

5

1

6

1

7

n j n

1 2 3 4 5 6 7

11

2

1

3

1

4

1

5

1

6

1

7

1

1

1

REMARKS:

Relaxing MCT: It is not necessary that the

sequence be monotonic initially, only that

they be monotonic from some point on,

that is, for n>K.

Two eventually similar sequences have

the same limit.