# Exam am ```BC 3
Sequences and Series Quiz
No Calculators allowed.
Show set-up/method clearly on all problems.
Name:
n2  3n
#1(4 pts) Given  ak  2
for all n  1, determine whether the series
3n  2n  2
k 1
converges or diverges and if it converges, find its sum.
n

#2 (2 pts each)
Suppose the series
a
n 1
n

a
k 1
k
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 bk  ak  0 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.
n!
3
n 1

b.
n2
 n 2  2n 

 3 2

n 5  3n  n  5 

c.
n
e
n 1
n
#3 (continued) Determine whether each series converges or diverges. Explain reasoning
carefully and completely.

d.
1
 n 3
n 1
n
n
1
approximates the value of the series
k 1 3  k !
#4 (6 pts). Find a value of n such that S n  

k
1
with an error of at most .001. Explain carefully. You need only find the value of
k!
k 1
n, you need not find the approximation S n for this value of n.
3
k

1
, where f n is the Fibonacci sequence defined by:
k 1 k  f k  2
f1  f 2  1, and f n  f n 1  f n  2 for n  3 .
#5(3 pts) Evaluate
f
```