1    Math 1432 Dr Almus  

Math 1432 Exam 4 Review

1. Determine if the following sequences are (i) monotonic, (ii) bounded.

a)2 10

12nn

an

b) 1

1n

nan

c) 14

12nn

an

12. Give the least upper bound and greatest lower bound (if they exist):

a) 5

12cos2n

na

b) 1

412

nn

c) 1

412

nn

2    Math 1432 Dr Almus  

2. Determine if the following sequences converge or diverge. If they converge, find the limit.

a) 2

4 1

1

n

n

b) 2

2

6 1

4 1

n n

n

d) 3ne

e)

54

1n

n

f) 3

ne

n

g) 5

!

n

n

h) 1809

!

n

n

i) 1

2

3

n

3    Math 1432 Dr Almus  

j) 1

!

4n

n

k) 2/nn

l) 1/5 nn

m)3

1

2

5

n

n

n) ln 2

5

n

n

o) 2 2ln 5 1 ln 4n n

4    Math 1432 Dr Almus  

3. Determine whether the given series converges or diverges; state which test you are using to determine convergence/divergence and show all work.

a. 4n2 1

n3 n

b. 4n2 1

n5 n

c. 2

( 1)!

k

k

d. 1

1

2

9

n

n

e. 2ln

5

n

n

f. 2n 1

n5 3n3 1

g. 11

n

n

h. 8 15

2

nn

n

i. 4 15

8 2

nn

n

j. 12

( 5)nn

5    Math 1432 Dr Almus  

k. n3

3n

l. 4

n

n

e

m. 42 n

nn

n. 3/2

2

5

2 1

n n

n n

o. 2

10 n

p. 210

n

n

n

q. 2

arctan( )

1

n

n

6    Math 1432 Dr Almus  

4. Determine if the following series (A) converge absolutely, (B) converge conditionally or (C) diverge.

a.

1

1

1

3

n

n

n

n

b. 1

1

1 5

3

n

n n

c. 1 7

91

1 1

3

n

n

n

n n

d. 1 2

1

1 ( 1)

( 5)( 6)

n

n

n

n n

e. 1

3 11

1 5

2

n n

nn

7    Math 1432 Dr Almus  

5. Determine whether the series converge or diverge; justify your answer. If convergent, find the sum (if possible).

a. 0

3

4

k

k

b. 2

12

9kk

k

c. 3 1

0

5

2

k

kk

d. 1

20

1 6

7

k

kk

8    Math 1432 Dr Almus  

6. Find the radius of convergence and interval of convergence for the following Power

series:

a. 0

( 4)

3

n

nn

x

n

b. 0

11

3

n

nn

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

9    Math 1432 Dr Almus  

 

7. Give the derivative of each power series below:

a.

02 2

)1(

n

n

n

xn

b.

0 12n

n

n

x

8. For each of the problems in number 4, give the antiderivative F of the power series so

that F(0)=0.

a.

02 2

)1(

n

n

n

xn

b.

0 12n

n

n

x

9. Use the Taylor series expansion (in powers of x) for f (x) ex to find the Taylor series

expansion f (x) cosh x .

10    Math 1432 Dr Almus  

10. Assume that f is a function such that for all n and x.

a. Estimate the maximum possible error if P4 (0.5) is used to approximate f (0.5)

b. Find the least integer n for which Pn (0.5) approximates f (0.5) with an error less

than 103.

11. Use the values in the table below and the formula for Taylor polynomials to give the 5th degree Taylor polynomial for f centered at x = 0.

f (0) f ' (0) f '' (0) f ''' (0) f (4) (0) f (5) (0)

1 0 -2 3 -4 1

12. Write the 8th degree Taylor Polynomials for:

a) 24( ) xf x e

b) 3 si (2 )( n)f x x x

c) ( ) cos(4 )f x x x

11    Math 1432 Dr Almus  

13. (a) Let 1

( )6

f xx

. Express this function as a power series using sigma notation.

(b) Let 2( )6

xg x

x

. Express this function as a power series using sigma notation

using your answer from part (a). (c) Express ( ) arctan(2 )h x x as a power series using integration. (not related to parts a

or b)

12    Math 1432 Dr Almus  

14. (a) Let 12

( )1 12

f xx

. Express this function as a power series using sigma notation.

(b) Let ( ) ln(1 12 )g x x . Express this function as a power series using sigma notation

using your answer from part (a).