MAE101 CAL v2 chapter 7 sequences and series

If the sequence {Sn} is convergent, then series is called convergent.. Example Find the nth partial sum of the following series and determine whether the series is convergent or diverg

Chapter 5

Sequences and Series

(Page 427-509, Calculus Volume 2)

Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn

Definition

A sequence is a list of numbers written in definite order:

a1, a2,…,an ,…

a1 is first term, a2 is second term,…, an is the nth term

Notation: The sequences {a1, a2,…,an,…} is also denoted by {an}

Example

Find the first 6 terms of Fibonacci sequence {f n }

2 1

2

1 = f = 1 , fn = fn− + fn−

f

1, 1, 2, 3, 5, 8, …

Theorem

n n

lim )

( lim )

(

n

n n

n n

lim /

) (

lim )

/ (

n

n n

n n

lim )

( lim )

(

n

n n

n n

Suppose that f :R→R such that limx→∞f(x) = L Then the

sequence {an = f(n)} also converges to L

( lim ( ) )

)) (

n n

sin

|

| 0

lim

r if

diverges

r if

r if

rn

n

Definition

A sequence {a n } is called increasing if a n < a n+1 for all n

A sequence {a n } is called a decreasing if a n > a n+1 for all n.

A sequence is called a monotonic if it is either increasing or

deceasing

Definition

A sequence {a n } is bounded above if there exists a number M

such that a n ≤ M for all n

A sequence {a n } is bounded below if there exists a number m

such that a n ≥ m for all n

A sequence is bounded if it is both bounded above and below.

Definition

The sum (infinite) of terms of a sequence {an}, denoted by

is called a series

The sum of first n terms:

is called the nth partial sum

If the sequence {Sn} is convergent, then series is called

convergent Otherwise, the series is called divergent.

1 2 1

Example

Find the nth partial sum of the following series and

determine whether the series is convergent or divergent.

1

2 )

n

n

n

n b

n a

5.2 Infinite Series

a ax

1 2

1 2

1

1

+ +

n

n n

5.2 Infinite Series

) (a n

Theorem

(a) If the series is convergent, then

(b) (Test for divergence)

If or does not exist, then the

5.3 The Divergence and Integral Tests

n n



=1 2 +

2

4 5

The Integral Test

Suppose that f(x) is a continuous, positive, decreasing on

[k, ∞) and let f(n)=a n Then:

• If converges, then also converges

• If diverges, then also diverges.

k

dx x

=k

n

n a





k

dx x

=k

n

n a

5.3 The Divergence and Integral Tests

Comparison Test

Suppose that and are non-negative series

Assume a n ≤ b n for all n ≥ k Then,

• If converges, then also converges

• if diverges, then diverges



=k n

n

b

5.4 Comparison test

5.4 Comparison test

n n

Compare the given series with

0 <c = 1, finite and converge →

5.4 Comparison test

= −

+2

Definition

The series is called an alternating series if its terms are

alternately positive and negative, that is,

1 3

1 2

1 1

1 )

1 (

) 1 (

n

n

n n



1

2 )

5.5 Alternating series

11

)1(

1

1

+

−+

Example

Is the series convergent?

=1

)cos(

n

5.5 Alternating series

If the series converges, then the series also

converges and we say that it converges absolutely.

A series is called conditionally convergent if it

is convergent but not absolutely convergent



=k n

n

a

5.5 Alternating series

| ) sin(

| )

sin(

n

n n

n

1 )

n

5.5 Alternating series

Show that the series is conditionally convergent

n

n

n

5.5 Alternating series

The Ratio Test (D’Alembert)

Given a series Let

Then,

• If L < 1, then the given series converges absolutely

(hence, converges)

• If L > 1, then the series is divergent.

•If L = 1, the Ratio Test is inconclusive; that is, no

conclusion can be drawn We have to use other tests

(e.g., integral, comparison, alternating series Tests)

n

n n

! ) 1 (

5 5

)!

1 (

) 1

( lim

1 1

n

n a

1

(

n

L > 1 → the given series is divergent

5.6 Ratio and Root test

5.6 Ratio and Root test

The Root Test (Cauchy)

Given a series Let

Then,

• If L < 1 then the given series converges absolutely

(hence, converges)

• If L > 1, then the given series diverges.

•If L = 1, the Root Test is inconclusive; that is, no

conclusion can be drawn We have to use other tests

(e.g.,the integral, comparison, alternating series Tests)

Is convergent?

= 1 +

2 13

n a

L

/ 1

2 1

/ 1

Guidelines for convergent/divergent testing.

•If limn→∞an≠0 the series ∑ an diverges.

•If an is of the form 1/np or xn, use the property of these special

series.

• If an is a rational fraction or roots of polynomials, use the

comparison/limit comparison Test.

•If in the expression of an has ! or xn , use the Ratio Test.

•If an = (bn)n, use the Root Test or Ratio Test.

•If an is alternating, use the alternating series test.

•If an = f(n) is positive and decreasing, use the Integral Test.