BC 3
Sequences and Series Quiz
No Calculators allowed.
Show set-up/method clearly on all problems.

#1 (2 pts each)
Suppose the series
 (1)
n 1
n 1
Name:
 an converges to 5 and an  an1  0 for
all n  1. Indicate whether each of the following statements is T (must be true) or F
(false). You may write a quick one sentence explanation, this may get you partial credit.
a.
The sequence ak must converge.
b.
The partial sum S101 could have a value of 5.01.
c.
The series

a
n 1

 (1)
d.
k  n 1
k 1
n
must converge.
 ak  an 1 for all n  1.

e.
If f (n)  an for each n  1 and


f ( x)dx converges, then  (1) n 1  an
1
n 1
converges absolutely.
(1) n
#2(6 pts) The series  n converges. (This can be shown using the Alternating Series
k 1 n
Test. But you need not show this). Find a value of n, such that the partial sum S n

(1) n
with an error less than .001.
n
k 1 n

approximates the value of 
#3(6 pts each) Determine whether each series converges or diverges. Explain reasoning
carefully and completely.
 n3  3n  1

 5
2 
n 1  4n  2n 

a.

b.
n 1
n
 5n (n !) 2 



n 1  (2n)! 

c.
n
3
#4(6 pts each) Determine whether each series converges absolutely, converges
conditionally, or diverges. Explain reasoning carefully and completely.

a.
 (1)
n
n 1

b.
 (1)
n 1
n
1
n2
1
 sin  
n
#5(6 pts) Find an expression for the sequence of partial sums, S n , for the series

 cos(k   ) . Use this to determine whether the series converges or diverges. If the series
k 1
converges, find the value.

#6(3 pts) Determine whether
k 1
carefully.
(2k )!
converges or diverges. Explain all steps
k
k!
k