9.5

Testing Convergence at Endpoints

Quick Review

413

2

1 3

1

1 2

Determine whether the improper integral converges or diverges.

11.

2. 1

ln3.

1 cos4.

dxxx

dxxxdx

xxdx

x

Quick Review

2

2

2

Determine whether the function is both positive and decreasing

on some interval ( , ).

5. ( ) 3 /

76. ( )

83

7. ( )3sin

8. ( )

N

f x x

xf x

xx

f xxx

f xx

Quick Review Solutions

413

2

1 3

1

1 2

conver

Determine

ges

whether the improp

diverges

diverges

conver

er integral converges or diverges.

11.

2. 1

ln3.

ges1 cos

4.

dxxx

dxxxdx

xxdx

x

Quick Review Solutions

2

2

2

Determine whether the function is both positive and decreasing

on some interval ( , ).

5. ( ) 3 /

76. ( )

83

7. ( ) 3sin

Yes

Y

8. ( )

es

No

No

N

f x x

xf x

xx

f xxx

f xx

What you’ll learn about Integral Test Harmonic Series and p-series Comparison Tests Alternating Series Absolute and Conditional Convergence Intervals of Convergence A Word of Caution

… and whyAdditional tests for convergence of series are introduced in

thissection.

The Integral Test

Let be a sequence of positive terms. Suppose that ( ),

where is a continuous, positive, decreasing function of for

all ( a positive integer). Then the series and the

integra

n n

nn N

a a f n

f x

x N N a

l ( ) either both converge or both diverge.N f x dx

Example Applying the Integral Test

1

1Does converge?

n n n

3 / 2 1/ 2

1 1 1

1The Integral Test applies because ( ) is a continuous, positive,

decreasing function of for 1.

1lim lim 2

2 lim 2 2

Since the integral con

kk

k k

k

f xx x

x x

dx x dx xx x

k

verges, so must the series.

Harmonic Series p-series

1

1 is called a , where is a real constant.

A is the -series where =1.

pnp

n

p p

series

harmonic series

p -

The Limit Comparison Test (LCT)Suppose that 0 and 0 for all ( a positive integer).

1. If lim , 0 , then and both converge or both diverge.

2. If lim 0 and converges, then converges.

3. I

n n

n

n nn

n

n

n nn

n

a b n N N

ac c a b

b

ab a

b

f lim and diverges, then diverges.n

n nn

n

ab a

b

Example Using the Limit Comparison Test

21

Determine whether the series converges or diverges.

3 5 7 2 1...

4 9 16 1n

n

n

2 2

2

2

2 1 2 2For large, behaves like , so compare it to the terms of (1/ ).

1

2 11 2 1

ˆlim lim lim 2 (Use l'Hopital's Rule)1 11

Since the limit is positive and (1/ ) diverg

n

n n n

n

n nn n

n nn

nna n n

b nn

n

21

2 1es, also diverges.

1n

n

n

The Alternating Series Test (Leibniz’s Theorem)

1

1 2 31

1

The series -1 ...

converges if all three of the following conditions are satisfied:

1. each is positive;

2. for all , for some integer ;

3. lim 0.

n

nn

n

n n

nn

u u u u

u

u u n N N

u

The Alternating Series Estimation Theorem

1

1

1

If the alternating series 1 satisfies the conditions of

The Alternating Series Test, then the truncation error for the

th partial sum is less than and has the same sign as the

first

n

nn

n

u

n u

unused term.

Rearrangement of Absolutely Convergent Series

1 2

1 1

If converges absolutely, and if , , ..., ,... is any rearrangement

of the sequence , then converges absolutely and

n n

n n n nn n

a b b b

a b b a

Rearrangement of Conditionally Convergent Series If converges conditionally, then the terms can be rearranged to

form a divergent series. The terms can also be rearranged to form a

series that converges to any preassigned sum.

na

How to Test a Power Series for Convergence

1. Use the Ratio Test for find the values of for which the series converges

absolutely. Ordinarily, this is an open interval - .

In some instances, the series converges for all values of .

x

a R x a R

x

In rare cases,

the series converges only at .

2. If the interval of absolute convergence is finite, test for convergence or

divergence at each endpoint. The Ratio Test fails at these points. Use a

c

x a

omparison test, the Integral test, or the Alternating Series Test.

3. If the interval of absolute convergence is - , conclude that

the series diverges for - , because for those values of th

a R x a R

x a R x

e th term

does not approach zero.

n

Example Finding Intervals of Convergence

2 2 4 61

1

For what values of does the following series converge?

-1 ...2 2 4 6

nn

n

x

x x x x

n

2 2

1

2

2 2

Apply the Ratio Test to find the interval of absolute convergence, then check

the endpoints if they exist.

2lim lim

2 2

2 lim

2 2The series converges absolutel

n

n

nn n

n

n

u x n

u n x

nx x

n

2

1

1

y for 1 or on the interval -1,1 .

-1At 1 and -1, the series is . This converges by the

2Alternating Series Test. The interval of convergence is -1,1 .

n

n

x

x xn

Procedure for Determining Convergence

Quick Quiz Sections 9.4 and 9.5

4

20 1 1

You may use a graphing calculator to solve the following problems.

1. Which of the following series converge?

2 2 1I. II. III.

1 3 1

(A) I only

(B) II only

(C) III only

(D) I

n

nn n n

n

n n

I and III

(E) I and II

Quick Quiz Sections 9.4 and 9.5

4

20 1 1

You may use a graphing calculator to solve the following problems.

1. Which of the following series converge?

2 2 1I. II. III.

1 3 1

(A) I only

(B) II only

(C) III only

(D) I

n

nn n n

n

n n

(E)

I an

I an

d III

d II

Quick Quiz Sections 9.4 and 9.5

1

2. Which of the following is the sum of the telescoping series

2 ?

1 2

(A) 1/3

(B) 1/2

(C) 3/5

(D) 2/3

(E) 1

n n n

Quick Quiz Sections 9.4 and 9.5

1

2. Which of the following is the sum of the telescoping series

2 ?

1 2

(A) 1/3

(B) 1/2

(C) 3/5

(D) 2/

(E) 1

3

n n n

Quick Quiz Sections 9.4 and 9.5

1

3. Which of the following describes the behavior of the series

ln 1 ?

I. converges II. diverges III. converges conditionally

(A) I only

(B) II only

(C) III only

(D) I and

n

n

n

n

III only

(E) II and III only

Quick Quiz Sections 9.4 and 9.5

1

3. Which of the following describes the behavior of the series

ln 1 ?

I. converges II. diverges III. converges conditionally

(A) I only

(B) II only

(C) III only

(D) I and

n

n

n

n

(E) II and

III

III

only

only

Pg. 386, 7.1 #1-25 odd