Chapter 9 Test
MAC 2312
Show all work for full or partial credit.
(3 points each unless otherwise noted)
_____ 1.
Name: ____________________________________
Match the sequence with its graph.
A)
B)
2n
an =
n!
C)
D)
Determine whether the following sequences converge or diverge.
If the sequence converges, specify to what number. (No justification is needed)
_____ 2.
_____ 3.
_____ 4.
Determine the convergence or divergence of the sequence.
A)
Converges to: ______________
B)
Diverges
Determine the convergence or divergence of the sequence.
A)
Converges to: ______________
B)
Diverges
Determine the convergence or divergence of the sequence.
A)
Converges to: ______________
B)
Diverges
1 
n
an = ( −1) 

 ln(n) 
an =
n2
2n 2 + n − 4
ln(n 2 )
an =
n
_____ 5.

Match the series with the graph of its sequence of partial sums.
A)
_____ 6.
7.
B)
4

 
n =0  5 
C)
D)
Which of the following would NOT prove that the series  an diverges?
A)
The sequence {an } diverges.
B)
0  bn  an and  bn diverges.
C)
lim an = 1
D)
The series  an diverges.
n →
Determine the sum of the convergent series:
8. Determine the sum of the convergent series.

1

 
n=2  2 
n
9. Determine the sum of the convergent series.

 2e
n=0
−n
1+
n
1 1 1
1
+ +
+
+ ...
3 9 27 81
Determine whether the following series converge or diverge. (No justification is needed)

_____ 10.
Determine the convergence or divergence of the series.
n =1
A)
Converges
B)
Diverges
_____ 11.
2

Determine the convergence or divergence of the series.
_____ 12.
n n
 n + n−5
A)
Converges
B)
Diverges
 2n 



n =1  3n + 1 

Determine the convergence or divergence of the series.
n
 tan (n)
−1
n =1
A)
Converges
B)
Diverges
Prove whether the following converge or diverge. Proper justification is needed for full credit,
which means that the convergence test used should be stated and properly applied.
(5 points each)
_____ 13. Prove the convergence or divergence of the series.

n2

3
n =1 n + 1
A)
Converges
B)
Diverges
_____ 14. Prove the convergence or divergence of the series.

en
n
n =1
A)
Converges
B)
Diverges
_____ 15. Prove the convergence or divergence of the series.

1
 n ln(n)
n=2
A)
Converges
B)
Diverges
_____ 16. Prove the convergence or divergence of the series.

n
A)
Converges
B)
Diverges
 n 



n = 2  ln( n ) 
_____ 17. Prove the convergence or divergence of the series.

cos 2 (n)

n2
n =1
A)
Converges
B)
Diverges
Prove whether the following converge absolutely, converge conditionally or diverge. Proper
justification is needed for full credit, which means that the convergence test used should be
stated and properly applied. (6 points each)
_____ 18. Prove whether the series converges absolutely, converges conditionally, or diverges.

 (−1)n
n =1
en
n!
A)
Converges Absolutely
B)
Converges Conditionally
C)
Diverges
_____ 19. Prove whether the series converges absolutely, converges conditionally, or diverges.
(−1)n

3
n
n =1

A)
Converges Absolutely
B)
Converges Conditionally
C)
Diverges
(5 points each)
U
20. Use the second degree Macluarin polynomial for
f ( x) = e x to approximate the value of e 0.1 .
21. Determine the interval of convergence of the power series. (Be sure to include a check for convergence at
the endpoints of the interval.)
(2 x + 3)n

n
n =1

22. A formula that is used extensively throughout mathematics and identifies an interesting mathematical
connection is Euler’s Formula:
eix = cos x + i sin x
where i = − 1
Use the power series expansions of these functions to verify this identity.