www.MasterMathMentor.com Stu Schwartz

A.P. Calculus 2 - Practice Exam. For each series, you are to determine whether it is convergent or divergent. You may do the problem using any test you wish, but you must identify the test you use and then justify your answer (example: p-series, n > 1 or geometric series, r > 1 or limit comparison to

!

1

n or ratio

test,

!

limn"#

an

<1, etc. The tests are: nth term, geometric, telescoping, p-series, alternating series, integral, root, ratio, direct comparison, limit comparison.

1.

!

n + 3

2n " 5n=1

#

$ 2.

!

n

n3

+1n=1

"

# 3.

!

n

n2

+ 3n=1

"

#

4.

!

3n

n3

n=1

"

# 5.

!

1.002( )n

n=1

"

# 6.

!

2n

3n +1

"

# $

%

& '

n=1

(

)2n

7.

!

"1( )n n

3n "1n=1

#

$ 8.

!

4n

n!n=1

"

# 9.

!

1

n7 6

n=1

"

#

www.MasterMathMentor.com Stu Schwartz

10.

!

1

nsin

2n "1( )#2

n=1

$

% 11.

!

1

n lnn( )2

n= 2

"

# 12.

!

4

2n "1n=1

#

$

13.

!

6

2n

n=1

"

# 14.

!

1

n"

1

n + 2

#

$ %

&

' (

n=1

)

* 15.

!

4n

5n

+1n=1

"

#

16.

!

"1( )n+1

nn=1

#

$ 17.

!

1

n2

+ 4n=1

"

# 18.

!

n3

4

"

# $ %

& '

n

n=1

(

)

19.

!

.9( )n

n=1

"

# 20.

!

n!

n2n

n=1

"

# 21.

!

1000

n7

n=1

"

#

www.MasterMathMentor.com Stu Schwartz

A.P. Calculus 2 - Practice Exam. Check your answers as to convergence and divergence. I show you the test I used and the rationale than I used to prove it. That does not mean you cannot use another test as long as you get the same answer as to convergence or divergence and your rationale is correct.

1.

!

n + 3

2n " 5n=1

#

$ 2.

!

n

n3

+1n=1

"

# 3.

!

n

n2

+ 3n=1

"

#

Divergent Convergent Divergent Test Used: nth-term Test Used: Limit Comparison Test Used: nth-term

Rationale:

!

limn"#

n + 3

2n + 3=1

2 Rationale:

!

limn"#

n

n3

+1$n2

1=1 Rationale:

!

limn"#

n

n2

+ 3=1

4.

!

3n

n3

n=1

"

# 5.

!

1.002( )n

n=1

"

# 6.

!

2n

3n +1

"

# $

%

& '

n=1

(

)2n

Divergent Divergent Convergent Test Used: Ratio Test Used: Geometric Test Used: Root

Rationale:

!

limn"#

3n+1

n +1( )3$n3

3n

= 3 Rationale:

!

r >1 Rationale:

!

limn"#

2n

3n +1

$

% &

'

( ) 2n*

+ ,

-

. /

1

n

=4

9

7.

!

"1( )n n

3n "1n=1

#

$ 8.

!

4n

n!n=1

"

# 9.

!

1

n7 6

n=1

"

#

Divergent Convergent Convergent Test Used: nth term Test Used: Ratio Test Used: p-series

Rationale:

!

limx"#

n

3n $1=1

3 Rationale:

!

limn"#

4n+1

n +1( )!$n!

4n

=

limn"#

4

n +1= 0

Rationale:

!

p >1

www.MasterMathMentor.com Stu Schwartz

10.

!

1

nsin

2n "1( )#2

n=1

$

% 11.

!

1

n lnn( )2

n= 2

"

# 12.

!

4

2n "1n=1

#

$

Convergent Convergent Divergent Test Used: Alternating Test Used: Integral Test Used: Integral

Rationale:

!

1"1

2+1

3+K+

1

n"

1

n +1 Rationale:

!

ln x( )"2

xdx =

2

#

$"1

ln x

%

& ' 2

#

Rationale:

!

4

2x +1dx =

1

"

# 2ln 2x $1( )]1

"

13.

!

6

2n

n=1

"

# 14.

!

1

n"

1

n + 2

#

$ %

&

' (

n=1

)

* 15.

!

4n

5n

+1n=1

"

#

Convergent Convergent Convergent Test Used: Geometric Test Used: Telescoping Test Used: Ratio

Rationale:

!

r <1 Rationale:

!

1"1

3+1

2"1

4+1

3"1

5K Rationale:

!

limn"#

4n+1

5n+1

+1$5n

+1

4n

=4

5

16.

!

"1( )n+1

nn=1

#

$ 17.

!

1

n2

+ 4n=1

"

# 18.

!

n3

4

"

# $ %

& '

n

n=1

(

)

Convergent Convergent Convergent Test Used: Alternating Test Used: Limit Comparison Test Used: Ratio

Rationale:

!

1

n +1<1

n Rationale:

!

limn"#

1

n2

+ 4$n2

1=1 Rationale:

!

limn"#

n +1( ) 34( )

n+1

n 34( )

n=

limn"#

3

4

$

% & '

( )

n

= 0

19.

!

.9( )n

n=1

"

# 20.

!

n!

n2n

n=1

"

# 21.

!

1000

n7

n=1

"

#

Convergent Divergent Divergent Test Used: Geometric Test Used: Ratio Test Used: p-series

Rationale:

!

r = .9 <1 Rationale:

!

limn"#

n +1( )!n +1( ) $ 2n+1

$n $ 2

n

n!

= limn"#

n

2=#

Rationale:

!

p <1