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

What’s 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):

𝑓 𝑛 + 2 = 𝑓 𝑛 + 1 + 𝑓 𝑛 , 𝑤ℎ𝑒𝑟𝑒 𝑓(1) = 1, 𝑓(2) = 1

General term (solution to the difference equation):

𝑓 𝑛 =5 + 5

10

1 + 5

2

𝑛−1

+5 − 5

10

1 − 5

2

𝑛−1

FYI: lim𝑛→∞

𝑓(𝑛+1)

𝑓(𝑛)= ϕ =

1+ 5

2= 1.618 … (golden ratio)

Recurrence formula

Explicit formula

OUR INTEREST IN SEQUENCES:

Behavior of f n

nas

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 2

1

e 3

1

e4

1

e 5

1

e 6

1

e 7

1

e

n

h n

1 2 3 4 5 6 7

1

e 2

1

e 3

1

e4

1

e 5

1

e 6

1

e 7

1

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

𝑛→∞1 −

1

𝑛= 1

For any real number 𝜖 > 0, take

𝑁 =1

𝜖

If 𝑛 > 𝑁, then

𝑛 >1

𝜖

𝜖 >1

𝑛

𝜖 > −1

𝑛

𝜖 > 1 −1

𝑛− 1 .

We need to find this Illustration: Suppose 𝜖 = 0.01

𝑁 =1

0.01= 100

Hence, for all 𝑛 > 100, the

distance 1 −1

𝑛− 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

11n

!

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. 11n

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

3

3

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 ½ 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.