Infinite Series

Chapter 9

9.1

Quick Review

 





 

1 2 1

Find the first four terms and the 30th term of the sequence , ,..., ,... .

4 1.

2 1 2.

The following sequences are geometric. Find the common ratio, the tenth term, and a r

n n n

n

n

n

u u u u

u

n u

n

  

   

(a)

(b) (c)





ule for the th term. 3. 2, 6, 18, 54,...

Quick Review





2

Determine lim . 1

5.

1 6. 1

ln 1 7.

1 8. 2

n n

n

n

n

n

n

a n

a

n a

n n a

n a

n



 

 

 _{ }

 

 

Quick Review Solutions

 





 

1 2 1

4/3, 1, 4/5,

Find the first four terms and the 30th term of the sequence , ,..., ,... .

4

1.

2 1 2

2/3, 1/8

-1, 1/2, -1/3, 1/4,

.

The following sequences are geometric. Fi 3

n 1

d / 0

n n n

n

n

n

u u u u

u

n u

n

  

   

(





 

the common ratio, the tenth term, and a rule for the th term.

3. 2, 6, 18, 54,... (a) 3 (b) 39,366 (c) 2 3 n

a

n





a)

Quick Review Solutions





2

Determine lim . 1

5.

1

6. 1

ln 1

7.

1 8. 2

0

0

2

n n

n

n

n

n

n

e

a n

a

n a

n n a

n a

n



 

 

 _{ }

 

 

What you’ll learn about



Geometric Series



Representing Functions by Series



Differentiation and Integration



Identifying a Series

… and why

Infinite Series

1 2 3 ₁

1 2

An is an expression of the form ... ..., or .

The numbers , ,... are the of the series; is the .

n _k k

n

a a a a a

a a a

 



    

infinite series

terms th term

Example

Identifying a Divergent Series

Does the series 2 2 2 2 2 2 ... converge?     

If this infinite series has a sum it has to be the limit of its partial sums, 2,0,2,0,2,... .

Example

Identifying a Convergent Series

7 7 7 7

Does the series ... ... converge? 10 100 1000   10n 

The sequence of partial sums, written in decimal form is 0.7, 0.77, 0.777, ...

Infinite Series

1 2 1

If the ... ...

converges, then lim 0.

k k

k

k k

a a a a

a

  

     



Geometric Series

1

The geometric series ... ...

converges to the sum if | | 1, and diverges if | | 1. (1 )

n n

n

a ar ar ar ar ar

a

r r

r



 





      

 

Example

Analyzing Geometric Series

1 1

Tell whether each series converges or diverges. If it converges, give its sum. 1

5 2

n

n

 



  _{ }

 

The first term is 5 and 1/ 2. The series converges to 5

10 1 (1/ 2)

a  r 

Example

Analyzing Geometric Series

1

Tell whether each series converges or diverges. If it converges, give its sum. 3

5

k

k

 

 _{ }

 

0

First term is (3/ 5) 1 and 3/ 5. The series converges to

1 5

. 1 (3/ 5) 2

a   r 

Power Series

















2 0 1 2 0

2

0 1 2

0

An expression of the form ... ... is a . An expression of the form

... ...

is a . T

n n

n n

n

n n

n n

n

c x c c x c x c x

c x a c c x a c x a c x a

 

 





     

         

power series centered at

power series centered at

x = 0

x = a he term





n n

c x a a



th term center

Example

Finding a Power Series by

Differentiation





2 3 2

Given that 1/(1 ) is represented by the power series 1 ... ... on the interval ( 1,1), find a power series to represent 1/ 1 .

n

x x x x x

x         













Notice that 1/ 1 is the derivative of 1/(1 ). To find the power series, differentiate both sides of the equation

1

1 ... ... . 1

1

1 ... ...

1 1

1 2 3 .

1

n

n

x x

x x x

x

d d

x x x

dx x dx

x x x           _{      }  _         1

Term-by-Term Differentiation

































2

0 1 2

0

1 2 1

1 2 3

0

If ( ) ... ...

converges for , then the series

2 3 ... ...,

obtained by differentiation the series for term by term, conv

n n

n n

n

n n

n n

n

f x c x a c c x a c x a c x a

x a R

nc x a c c x a c x a nc x a

f

 

  







          

 

         

erges for

and represents '( ) on that interval. If the series for converges for all , then so does the series for '.

x a R f x f

x f

Example

Finding a Power Series by

Integration

 

2 3

1

Given that 1 ... ..., 1 1,

1

find a power series to represent ln(1 ).

n

x x x x x

x x             

 

 

 





Recall that 1/(1 ) is the derivative of ln(1 ). Integrate the series 1/(1 ) to find the series for ln(1 ).

1

1 ... ...

1

=1 ... 1 ... 1

1 ... 1 ...

1

n

n _n

n

x n

x x x

x

x x x x

x

x x x x

dt t t t t

t                              







 

 

2 3 4 1

0

0

2 3 4 1

ln 1 ... 1 ...

2 3 4 1

ln(1 ) ... 1 ...

x x n x n n n dt

t t t t

t t

n

x x x x

Term-by-Term Integration





















_

_













2

0 1 2

0

1 2 3 1

0 1 2

0

If ( ) ... ...

converges for - , then the series

... ...,

1 2 3 1

obtained by integrating the series for term by term,

n n

n n

n

n n

n n

n

f x c x a c c x a c x a c x a

x a R

x a x a x a x a

c c x a c c c

n n

f

 

 

 





          



   

      

 

converges for - and represents ( ) on that interval. If the series for converges for all , then so does the series for the integral.

x a

x a R f t dt f

x



9.2

Quick Review

 

_

_

3

2 1

Find a formula for the th derivative of the function. 1.

2. 3 3. 4. ln

Find .

5.

!

6. -1

2 1 !

x

x

n

n

n n

n e

x x

dy dx x y

n

x y

n



 

Quick Review Solutions





  







 

_

_

 

_{ }





Find a formula for the th derivative of the function. 1.

2. 3 3. 4. ln

3

3 ln 3 !

-1 1 !

1 !

1 2 !

Find .

5. !

6. -1

What you’ll learn about



Constructing a Series



Series for sin

x

and cos

x



Beauty Bare



Maclaurin and Taylor Series



Combining Taylor Series



Table of Maclaurin Series

… and why

Example

Constructing a Power Series for

sin

x

Construct the seventh order Taylor polynomial and the Taylor series for sin at =0.x x

    4 5 2 3 7 sin(0) 0

sin'(0) cos(0) 1 sin''(0) sin(0) 0 sin'''(0) cos(0) 1 sin (0) sin(0) 0

sin (0) cos(0) 1 The pattern 0, 1, 0, -1 repeats forever. The seventh order Taylor polynomial at 0:

1 ( ) 0 1 0

3!

x

P x x x x

                  

 

_

_

4 5 6 7

3 5 7

2 1

3 5 7 9

0

1 1

0 0

5! 7! 1 1 1

. 3! 5! 7!

1 1 1 1

The Taylor series: ... 1 . 3! 5! 7! 9! 2 1 !

n n

n

x x x x

x x x x

x

x x x x x

Taylor Series Generated by

f

at

x=0

(Maclaurin Series)

   

2

0

Let be a function with derivatives of all orders throughout some open interval

containing 0. Then the is

''(0) (0) (0)

(0) '(0) ... ... .

2! ! !

Th

n k

n k

k

f

f f f

f f x x x x

n k

 



     

Taylor series generated by at f x = 0

  0

is series is also called the .

(0)

The partial sum ( ) is the !

k n

k

n _k

f

P x x

k







Maclaurin series generated by

Taylor polynomial of order for

at 0

f

n

Example

Approximating a Function near

0

Find the fourth order Taylor polynomial that approximates cos 2 near 0.

y  x x 

 

 

   

_{   }

 

 

   

2 4 2

2 4 2

2 4

2 4

4

We know that cos 1 ... 1 ... .

2! 4! 2 !

2 2 2

Therefore, cos 2 1 ... 1 ... .

2! 4! 2 !

2 2

So, 1 1 2 (2 / 3) .

2! 4!

n n

n n

x x x

x

n

x x x

x

n

x x

P x x x

      

      

Taylor Series Generated by

f

at

x=a

















Let be a function with derivatives of all orders throughout some open

interval containing . Then the is

''( ) ( ) ( )

( ) '( ) ... ...

2! ! !

n k

n

f

a

f a f a f a

f a f a x a x a x a x a

n k

         

Taylor series generated by at f x = a

 





0

0

. ( )

The partial sum ( ) is the

!

k

k

k

n _k

n _k

f a

P x x a

k

 







  Taylor polynomial of order for at

Example

A Taylor Series at

x

= 1

Find the Taylor series generated by ( ) x at 1.

f x  e x 

 

















2

0

Observe that (1) '(1) ''(1) ... (1) .

Therefore, the series is 1 1 ... 1 ...

2! !

1 . !

n

n x

k

k

f f f f e

e e

e e e x x x

n e

x k

 



    

        

 

 _{ } 

Example

A Taylor Polynomial for a

Polynomial

3 2

Find the third order Taylor polynomial for ( )f x  x 3x  x 1 at x  0. This polynomial is already written in powers of and is

of degree three, so it is its own third order Taylor polynomial at 0.

x

Example

A Taylor Polynomial for a

Polynomial

3 2

Find the third order Taylor polynomial for ( )f x  x 3x  x 1 at x 1.

















3 2

1 2

1 1

2 3

3

3 2

(1) 3 1 4

'(1) 3 6 1 8 ''(1) 6 6 12 '''(1) 6

12 6

So, ( ) 4 8( 1) 1 1

2! 3!

1 6 1 8 4

x

x

x

f x x x

f x x

f x

f

P x x x x

x x x

  

    

   

  



      

Maclaurin Series





 

 









 

_

_

 

_

_





3 5 2 1 2 1

0

1

1 ... ... 1

1 1

1 ... 1 ... 1 1

1

1 ... ... all real

2! ! !

sin ... 1 ... 1 all real

3! 5! 2 1 ! 2 1 !

c

n n

n

n _n n _n

n n n x n n n n n n

x x x x x

x

x x x x x

x

x x x

e x x

n n

x x x x

x x x

n n                                                    

 

_{ }

 

_{ }





2 4 2 2

0

os 1 ... 1 ... 1 all real

2! 4! 2 ! 2 !

n n

n n

n

x x x x

Maclaurin Series

_

_

_{ }

_{ }

_

_

 

 





2 3

1 1

1

3 5 2 1 2 1

1

0

ln 1 ... 1 ... 1 1 1

2 3

tan ... 1 ... 1 1

3 5 2 1 2 1

n n

n n

n

n n

n n

n

x x x x

x x x

n n

x x x x

x x x

n n



 



 

 







            

         

9.3

Quick Review





 





2

Find the smallest number that bounds from above on the interval (that is, find the smallest such that ( ) for all in ).

1. ( ) 2cos(3 ), -2 , 2 2. ( ) 3 1, 2

3. ( ) 2 -3,0

4.

x

M f

I M f x M

x I

f x x I

f x x I

f x I

 



 

  

 





2

( ) -2, 2 1

x

f x I

x

 

Quick Review

2

-3 2

Tell whether the function has derivatives of all orders at the given values of .

5. , 0 1

6. 4 , 2 7. sin cos , 8. , 0

9. , 0

x

a x

a x

x a

x x a

e a

x a



 

 

 

Quick Review Solutions





 

2

Find the smallest number that bounds from above on the interval (that is, find the smallest such that ( ) for all in ).

1. ( ) 2cos(3 ), -2 , 2 2. ( ) 3 1, 2

3. ( ) 2

2 7

x

M f

I M f x M

x I

f x x I

f x x I

f x

 



 

  











2

-3,0

4. ( ) -2, 2 1

1

1/2 I

x

f x I

x



 

Quick Review Solutions

2

-3 2

Tell whether the function has derivatives of all orders at the given values of .

5. , 0 1

6. 4 , 2 7. sin co

Yes No

Y

s , 8. , 0

9. , 0

es Yes

No

x

a x

a x

x a

x x a

e a

x a



 

 

 

What you’ll learn about



Taylor Polynomials



The Remainder



Remainder Estimation Theorem



Euler’s Formula

… and why

Example

Approximating a Function to

Specifications





Find a Taylor polynomial that will serve as an adequate substitute for sin on the interval - , .

x  





Choose ( ) so that ( ) sin 0.0001 for every in the interval , . We need to make ( ) sin 0.0001, because then will be adequate throughout the interval.

( ) sin 0.0001 ( ) 0.0001

Evalua

n n

n n

n

n

P x P x x x

P P

P P

   

 



 

  

 



te partial sums at , adding one term at a time. Eventually you will find the following:

Taylor’s Theorem with Remainder













 





2

1

If has derivatives of all orders in an open interval containing , then for each positive integer and for each in

''( ) ( )

( ) ( ) '( ) ... ( ),

2! !

( ) where ( )

1 !

n

n

n

n

n

f I a

n x I

f a f a

f x f a f a x a x a x a R x

n

f c

R x

n



        









1

for some between and .

n

x a  c a x

Example

Proving Convergence of a

Maclaurin Series

 

_

_

0

Prove that the series -1 converges to sin for all real . 2 1 !

k k k x x x k       









Consider ( ) as . By Taylor's Theorem, ( ) 0 , 1 !

where ( ) is the ( 1)st derivative of sin evaluated at some between and 0. For this function -1 ( ) 1 so,

( ( ) n n n n n n n n f c

R x n R x x

n

f c n x c x

f c f R x              

















As , the factorial growth is larger in the

1 ! 1 !

denominator, than the exponential growth in the numerator. Therefore, as

Remainder Estimation Theorem

 





1 1

1 1

If there are positive constants and such that ( ) for all between and , then the remainder ( ) in Taylor's Theorem

satisfies the inequality ( ) . 1 ! If these conditio

n n

n

n n

n

M r f t Mr t

a x R x

r x a

R x M

n

 

 

 





ns hold for every and all the other conditions of Taylor's Theorem are satisfied by , then the series converges to ( ).

n

Example

Proving Convergence

0

Use the Remainder Estimation Theorem to prove that for all real . !

k x

k

x

e x

k

 





 1 1

We have already shown this to be the Taylor series generated by at 0.

We must verify ( ) 0 for all . To do this, we must find and such that ( ) is bounded by for between 0 an

x

n

n t n

e x

R x x M r

f  t e Mr  t

 



0

d arbitrary . Let be the maximum value of and let 1.

If the interval is [0, ], let .

If the interval is [ ,0], let 1.

In either case, throughout the interval,and the Remainder Es

t

x

t

x

M e r

x M e

x M e

e M

 

 

 timation

Euler’s Formula

cos sin

ix

Quick Quiz

Sections 9.1-9.3

2 0

2 2 2

1. Which of the following is the sum of the series ?

(A)

-(B)

-(C) -e

(D)

-(E) The series diverges

n

n

n e

e e

e

e e



  

 



 

Quick Quiz

Sections 9.1-9.3

2 0

2

2 2

1. Which of the following is the sum of the series ?

(

(

A)

-(B)

-(C) -e

(E) The series diverges

D)

-n

n

n e

e

e e

e

e



  

 



 

Quick Quiz

Sections 9.1-9.3

2

2. Assume that has derivatives of all orders for all real numbers , (0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following is the third order Taylor polynomial for at 0?

(A) 2 3

f x

f f f f

f x

x x

   



  3

2 3

2 3

2 3 2

2 (B) 2 6 12

1

(C) 2 3 2

2

(D) 2 3 2

(E) 2 6

x

x x x

x x x

x x x

x x



  

  

   

Quick Quiz

Sections 9.1-9.3

2

2. Assume that has derivatives of all orders for all real numbers , (0) 2, '(0) -1, ''(0) 6, and '''(0) 12. Which of the following is the third order Taylor polynomial for at 0?

( ) A 2 3

f x

f f f f

f

x

x

x

   



 

2 3

2 3

2

3

3 2

(B) 2 6 12 1

(C) 2 3 2

2

(D) 2 3 2

(E) 2

2

6

x x x

x x x

x x x

x

x

x

  

  

   

 

Quick Quiz

Sections 9.1-9.3





 

  



  



  



3. Which of the following is the Taylor series generated by ( ) 1/ at 1?

(A) 1

(B) 1

(C) 1 1

1 (D) 1

!

(E) 1 1

n n n _n n n n n n n n n n n

f x x x

Quick Quiz

Sections 9.1-9.3





 

  



  



  



3. Which of the following is the Taylor series generated by ( ) 1/ at 1?

(A) 1

(B) 1

(C) 1 1

1 (D) 1

!

(E) 1 1

n n n _n n n n n n n n n n n

f x x x

9.4

Quick Review









 

2

4 ₂ 4

1 1

Find the limit of the expression as . Assume remains fixed as changes.

1.

1 3 2.

1

3. ! n+1 4.

2

2 2 1 5.

2 2 1

n n

n n

n x n

n x n n x n n

x n

x n x x

 

  

 

Quick Review

2 5

Let be the th term of the first series and the th term of the second series. Find the smallest positive integer for which

for all . Identify and . 6. 5 ,

7. , 5 8.

n n

n n n n

n

a n b n

N

a b n N a b

n n

n

 

 

 

-3

ln , 1 9. ,

10n

n n

n

 

Quick Review Solutions









 

Find the limit of the expression as . Assume remains fixed as changes.

1. 1 3 2. 1 3. ! n+1 4. 2

2 2 1

5.

| |

| 3 |

0

/16

| 2 1| /

1 2 2 2 n n n n x

n x n

Quick Review Solutions

2 2

Let be the th term of the first series and the th term of the second series. Find the smallest positive integer for which

for all . Identify an

, 5

d .

6. 5 , ,

n n

n n n n

n n

a n b n

N

a b n N a b

a n b n

n n       - 3 2 3 5 5 2

7. , 5

8. ln ,

1 1

9. , 1

6 5 , , 6

, ln , 1

1 1

, ,

0 n! 10 ! 25

1

, , 2

1

10. ,

n

n n

n n

n n n n

n

n

n

N

a b n N

a n b n N

a b N

n

a b n

n

n n

n n n N

What you’ll learn about



Convergence



n

th-Term Test



Comparing Nonnegative Series



Ratio Test



Endpoint Convergence

… and why

It is important to develop a strategy for finding the

The Convergence Theorem for Power

Series

_

_

0

There are three possibilities for with respect to convergence: 1. There is a positive number such that the series diverges for

but converges for . The series may or may not con

n n

n c x a

R x a R

x a R

 

 

 

  verge at either

of the endpoints amd . 2. The series converges for every ( ).

3. The series converges at and diverges elsewhere ( 0).

x a R x a R

x R

x a R

   

 

The

n

th-Term Test for Divergence

1 n diverges if lim fails to exist or is different from zero.n n

n a a



 

The Direct Comparison Test

converges if there is a convergent series with for all , for some integer .

diverges if there is a divergent series with

n

n n

n n

n n

n n

a

a c

a c n N N

a d

a d



 

 

 



(a)

(b)

Example

Proving Convergence by

Comparison

 

Prove that converges for all real . ! n n x x n   

 

 

 

 

Let be any real number. The series has no negative terms. !

For any , . Recognize as the Taylor series

! ! !

!

for , which we know converges to for all real nu

n n n n n n n x x x x n x x x x n

n n n

n e e        

 

mbers. Since

the series dominates term by term, the latter series must also !

converge for all real numbers by the Direct Comparison Test.

Absolute Convergence

If the series of absolute values converges, then .

n n

a a

 

Absolute Convergence Implies

Convergence

Example

Using Absolute Convergence





0

sin

Show that converges for all . ! n n x x n    0 0 0 sin

Let be any real number. The series has no negative terms, !

1

and it is term-by-term less than or equal to the series , which converges !

sin

to . Therefore, converges by di ! n n n n n x x n n x e n         





The Ratio Test

Then,

the series converges if 1, the series diverges if 1, the test is inconclusive if 1.

n

n _n

n

a

a L

a

L L

L

 

 

 



Example

Finding the Radius of

Convergence

0

Find the radius of convergence of . 10 n n n nx   





1

1

Check for absolute convergence using the Ratio Test.

1 ₁₀ lim lim 10 1 lim 10 10

Setting 1, we see that the series converges absolutely for 10 10. 10 Th n _n n n n n n n n n x a

a n x

x x n n x x              _{ }       

Example

Determining Convergence of a

Series

0

2 Determine the convergence or divergence of the series .

3 1 n n n     1 1 1 1 1 1

Use the Ratio Test: 2

2 3 1

3 1

lim lim lim

2 _{3 1 2}

3 1 3 1 lim 2 3 1 1 1 3 lim 2 1 3 3 2 2

The series converges because 3

n

n n

n n

n _{n n}

n n n

9.5

Quick Review

4 1

3 2 1 ₃

1

1 ₂

Determine whether the improper integral converges or diverges. 1

1.

2.

1 ln 3.

1 cos 4.

dx x

x

dx x

x dx x

x dx x



















Quick Review

2 2 2

Determine whether the function is both positive and decreasing on some interval ( , ).

5. ( ) 3/ 7 6. ( )

8 3

7. ( ) 3 sin 8. ( )

N

f x x

x f x

x x f x

x x f x

x

 



  

Quick Review Solutions

4 1

3 2 1 ₃

1

1 ₂

conver

Determine

ges

whether the improp

diverges

diverges

conver

er integral converges or diverges. 1

1.

2. 1

ln

3.

ges

1 cos

4.

dx x

x

dx x

x dx x

x dx x



















Quick Review Solutions

2 2 2

Determine whether the function is both positive and decreasing on some interval ( , ).

5. ( ) 3/ 7

6. ( ) 8 3

7. ( ) 3

sin

Yes Y

8. ( )

es

No

No N

f x x

x f x

x x f x

x x f x

x

 



  

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 why

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

n n N

a a f n

f x

x N N  a





 

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



Example

Applying the Integral Test

1

Does converge?

n n n

 







3 / 2 1/ 2

1 1 ₁

1

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

decreasing function of for 1. 1

lim lim 2

2

lim 2 2

Since the integral con

k k

k k

k

f x

x x

x x

dx x dx x

x x

k

  

 



 

 

  

 

 _  _ 

 

Harmonic Series

p

-series

1

1

is called a , where is a real constant.

A is the -series where =1.

p

n n p

p p

 

_ _

  series

harmonic series

-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 n

n

n

n

n n

n

n

a b n N N

a

c c a b

b a

b a

b





 

 

  

   



f lim n and diverges, then diverges.

n n

n

n

a

b a

b

Example

Using the Limit Comparison

Test





Determine whether the series converges or diverges.

3 5 7 2 1

...

4 9 16 n 1

n n         

















2 1 2 2

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

2 1

1 2 1

ˆ

lim lim lim 2 (Use l'Hopital's Rule)

1 _{1 1}

Since the limit is positive and (1/ ) diverg

n

n n n

n n n n n n n n n n

a n n

b n n n       _    _     





2 1

The Alternating Series Test

(Leibniz’s Theorem)

 

1 2 3 1

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 n n

n

n n

n n

u u u u

u

u u n N N

u

 _ 

 

    

 

The Alternating Series Estimation Theorem

 

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 n n

n

u

n u

 _ 



 

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 n _n 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.

n

a

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.

Example

Finding Intervals of

Convergence

 

1

For what values of does the following series converge?

-1 ...

2 2 4 6

n n

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.

2 lim lim 2 2 2 lim 2 2

The series converges absolutel

n n n n n n n

u x n

u n x

n x x n            





 

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

x



Quick Quiz

Sections 9.4 and 9.5

4 2

0 1 1

You may use a graphing calculator to solve the following problems. 1. Which of the following series converge?

2 2 1

I. II. III.

1 3 1

(A) I only (B) II only (C) III only (D) I

n

n

n n n

n

n n

  

  

   

 

Quick Quiz

Sections 9.4 and 9.5

4 2

0 1 1

You may use a graphing calculator to solve the following problems. 1. Which of the following series converge?

2 2 1

I. II. III.

1 3 1

(A) I only (B) II only (C) III only (D) I

n

n

n n n

n

n n

  

  

   

 

(E)

I an

I an

d III

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

 

 

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

Chapter Test

 

0

Find the radius of convergence for the series and its interval of convergence. Then identify the values of for which the series converges absolutely and conditionally.

1.

!

2.

n

n

x

x n

 

 

(a) (b)

(c) (d)



 







0

1 2 1 2 1 2

3. The series is the value of the Maclaurin series of a function ( ) at a particular point. What function and what point? What is the

n

n n

n n

n

f x

 

  

Chapter Test

2

-4. Find a Maclaurin series for the function .

5. Find the first four nonzero terms and the general term of the Taylor series generated by at .

1

( ) , 2

3

1 6. Find the sum of the series

2

x

xe

f x a

f x a

x



 





 



0 3 2 1 .

n n n

 



Chapter Test

 





  



Determine if the series converges absolutely, converges conditionally, or diverges. 1 7. ln 1 1 1 8. 2 1

9. Let ( ) for all for which the series converges. !

(a) Find the

n n n n n n n n n n n x n

f x x

n                

radius of convergence of this series.

(b) Use the first three terms of this series to approximate (-1/ 3). (c) Estimate the error involved in the approximation in part (b).

Chapter Test

10. Let be a function that has derivatives of all orders for all real numbers. Assume that (3) 1, '(3) 4, ''(3) 6, and '''(3) 12.

(a) Write the third order Taylor polynomial for at 3 and use i

f

f f f f

f x

   

 t to

approximate (3.2).

(b) Write the second order Taylor polynomial for 'at 3 and use it to approximate '(2.7).

(c) Does the linearization of underestimate or overestimate the values of ( ) near

f

f x

f

f f x



3?

Chapter Test Solutions

 

0

Find the radius of convergence for the series and its interval of convergence. Then identify the values of for which the series converges absolutely and conditionally.

1. ! n n x x n     (a) (b)

(c) (d)



 















0

(a) (b) all real numbers (c) all real numbers (d) none

(a) 1 (b) -3/ 2, 1/ 2 (

1 2 1

2.

2 1 2

3. The series is the value of the Maclaurin series of a function ( ) at a

c) -3/ 2, 1/ 2 (d) n

p e ar on n n n n n n f x       

 

3 5 2 1

ticular point. What function and what point? What is the

sum of the series? ... 1 +...

n n

 



Chapter Test Solutions

 

2

5 7 2 1

3

-4. Find a Maclaurin series for the function .

5. Find the first four nonzero terms and the general term of the Taylor series generat

... 1 .

ed by at . 1

.. 2! 3

( ) , 2

3 ! ! n x n xe

f x a

f x a

x x x

x n x x            



 

 









 



1 2 2 2 ... 2 ...

-1/

es .

2

6

3 2 1

n

n

x x x

Chapter Test Solutions

 





  



Determine if the series converges absolutely, converges conditionally, or diverges. 1 7. ln 1 1 1 8. 2 1

9. Let ( ) for all for wh

converges conditionally diverges ! n n n n n n n n n n n x n

f x x

n               

 ich the series converges. (a) Find the radius of convergence of this series.

(b) Use the first three terms of this series to approximate (-1/ 3). (c) Estimate the error involved in th

1/

-5/18

e approx

e

f

Chapter Test Solutions

10. Let be a function that has derivatives of all orders for all real numbers. Assume that (3) 1, '(3) 4, ''(3) 6, and '''(3) 12.

(a) Write the third order Taylor polynomial for at 3 and use i

f

f f f f

f x

   























2 3

3

2

2 2

t to approximate (3.2).

(b) Write the second order Taylor polynomial for 'at 3 and use it to approximate '(2

( ) 1 4 3 3 3 2 3 ; (3.2) (3.2) 1.936

( ) 4 6 3 6 3 ; '(2

.7).

.7) (2.7) 2

P x x x x f P

P x x x f P

f

f x

f

        

      



.74 underestimat

(c) Does the linearization of or overestimate the values of ( ) near

e

3?

f