BC 3
Sequences and Series Quiz
No Calculators allowed.
Show set-up/method clearly on all problems.
Name:
#1(6 pts) Find an expression for the sequence of partial sums, S n , for the series

1
1 
  k  k  2  . Use this to determine whether the series converges or diverges. If the
k 1
series converges, find the value.

#2 (2 pts each)
Suppose the series
a
n 1
n
converges to 5 and an  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 S100 could have a value of 5.01.
c.
The sequence of partial sums must converge to 5.
d.
If the ratio test is applied to this series, then the value of lim
ak 1
k  ak
must be a finite positive number strictly less than 1.
e.
If ak  bk for all k  1 , then

b
n 1
n
must converge.
#3(6 pts each) Determine whether each series converges or diverges. Explain reasoning
carefully and completely.

a.
2n

n 1 n !

b.
n 1
n
 n 2  2n  3 

 3

2
n 1  3n  n  n 

c.
1
 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 
n 1
1
n
2n  n !
(2n)!

#5(6 pts) The series
(1) n
 n2
k 1
n
converges. (This can be shown using the Alternating Series
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  2

approximates the value of 

#6(3 pts) Determine whether
n 1
carefully and completely.
n! nn
 (2n)! converges or diverges. Explain reasoning