This document contains 5 practice problems for determining the convergence of infinite series using various tests. The tests covered are direct comparison, alternating series, ratio, limit comparison, and root tests. Students are asked to use these convergence tests to analyze the convergence of 5 given infinite series.

1. ECONOMICS 201: Introduction to Microeconomics
NUMBER ONE {60 points}. Using the production possibilities
frontiers for WOMEN and
MEN provided below, explain which group has a comparative
advantage over which type of
goods. Is it rational for WOMEN to trade with MEN? Explain.
What will keep each group
from trying to overcharge the other for their specialized
product? What are some of the barriers
to trade which might keep the two groups apart even if there are
mutually beneficial gains from
exchange possible?
Child Care (CC) Income Earning (IE)
WOMEN 100 400
MEN 80 240
NUMBER TWO {60 points}. Suppose there is market for
airfares to Caribbean Beach Vacation
Destinations (CBVDs). Consider the following list of factors
which might influence the market
for CBVDs:
A. the public becomes more workoriented
and less leisureoriented
B. the income of the public decreases
C. the prices of hotel rooms in the Caribbean beach vacation
regions increases slightly
D. the prices of airfares to Colorado ski vacation destinations
increases
E. the sales tax imposed on the purchase of airfares increases
substantially
F. the cost of jet fuel increases
G. the wages of travel agents decrease
Use the Supply and Demand Model to provide an answer for
each of the four complex cases
described below. What happens in the market for Caribbean

2. Beach Vacation Destinations if:
I. A and F occur at the same time
II. B and G occur at the same time
III. C and E occur at the same time
IV. C and D and F occur at the same time
NUMBER THREE {605 points}. The behavioral model of
neoclassical microeconomics asserts
that human beings are “selfinterestedand rational.” Yet, we have
emphasized the importance of
a human being’s social nature, and especially the importance of
acquiring morals and ethics,
essentially from our social settings. When you ask people about
what is important to them, most
people mention “family” somewhere near the top of their
priorities; rarely is a person so candid
as to admit that they place themselves first.
Is it important to be committed primarily to yourself or to your
family? What if someone in
your family breaks the law, or commits a fundamentally
immoral act? According to
microeconomics, how does a person decide where to give their
loyalty when the interests of one
social group, like the family, are in conflict with the interests of
another social group, like the
person’s nation or religion? In summary, what is the economic
role of ethics and morals for both
the individual person and the society?
NUMBER FOUR {60 points}. Consider the data on housing
prices provided below. Use the
Supply and Demand model to describe some possible
components of these changes in housing
prices.
Median Sales Price of Existing SingleFamily
Homes for Metropolitan Areas. [source: National
Association of Realtors.]
2012 2013 2014

3. USA Average Metro Area 177.2 197.4 208.9
Atlantic City NJ 220.4 218.3 207.6
BloomingtonNormal
IL 156.4 154.0 156.0
Cumberland MDWV
96.2 102.2 92.0
Los AngelesLong
BeachSanta
Ana CA 327.5 405.6 449.5
PortlandVancouverBeaverton
ORWA
232.9 266.5 286.0
San DiegoCarlsbadSan
Marcos CA 385.5 466.3 497.9
San FranciscoOaklandFremont
CA 543.8 669.6 737.6
SeattleTacomaBellevue
WA 300.4 336.3 356.6
NUMBER FIVE {60 points}.
In 1985, the Oregon Legislature passed a law which reads:
ORS 91.225 (1)
The Legislative Assembly finds that there is a social and
economic need to insure an adequate
supply of affordable housing for Oregonians. The Legislative
Assembly also finds that the
imposition of general restrictions on housing rents will disrupt
an orderly housing market,
increase deferred maintenance of existing housing stock, lead to
abandonment of existing rental
units and create a property tax shift from rentalowned
to owner occupied
housing. Therefore, the Legislative Assembly declares that the
imposition of rent control on housing in the State of Oregon is a
matter of statewide concern. Recently, the price charged for
rental apartments in Portland OR has been increasing, one
report offered the figure of an average 7.0% increase from last

4. year. Some political pressure is being generated currently to
repeal the 1985 State Law and to impose Rent Control Laws in
Portland. Use the Supply and Demand model to describe what
would happen in the market for rental apartments if such a price
ceiling scheme were imposed in Portland.
Infinite Series Convergence Tests
n
a , convergence can be evaluated by using the following tests.
Remember, the Geometric and Telescoping Series are the only
types of series whose convergence can be calculated exactly.
The other tests serve the purpose of determining whether or not
the series converges, not to what finite number it converges.
Test Format Results

5. 1
1
n
n
verges to
r
a
a
Expand an by Partial Fraction Decomposition
Canceling may occur, making the convergence obvious by
examining
n
n
s

6. lim as a
sequence, where sn = a1 + a2 + . . . + an.
p - series
p
n
a
If p > 1, the series converges.
1
)( dxxf , where f(n) = an for every n = 1, 2, 3 . . .
If the improper integral converges, so does the series.
If the improper integral diverges, so does the series.

7. n
n
b .
n
n
a .
n
b is

8. n
a .
Limit Comparison
n
n
b ,
by simplifying the ratio
n
n
b
a
If
n

9. n
n b
a
lim converges to a finite positive number a
n
n
b
either both converge or both diverge.
Infinite Series Convergence Tests

10. n
a , convergence can be evaluated by using the following tests.
Remember, the Geometric and Telescoping Series are the only
types of series whose convergence can be calculated exactly.
The other tests serve the purpose of determining whether or not
the series converges, not to what finite number it converges.
Test Format Results
1
1
)1(
n
n
n

11. If the following two criteria are met, then the series converges:
n
n
b
lim = 0 as a sequence.
Ratio Constuct the ratio
n
n
a
a
If
n
n
n a
a

12. 1
lim
< 1 as a sequence, then the series converges.
If
n
n
n a
a
1
lim
> 1 or diverges as a sequence, then the series diverges.
If
n
n
n a
a
1

13. lim
= 1 as a sequence, then the test fails.
Root Construct the quantity n na
If n
n
n
a
lim < 1 as a sequence, then the series converges.
If n
n
n
a
lim > 1 or diverges as a sequence, then the series diverges.
If n
n
n
a
lim = 1 as a sequence, then the test fails.

14. Math 210 – Fall 2015 Name: _______________________
Assignment: G10
Determine the convergence of these series.
4
n
n
n
(Direct Comparison)
(2)

15. 1
2
1
5
1
n
n
n
(Alternating)
2

16. 12n n
n
(Ratio or Integral?)
n
n
(Ratio)
1 5
12
n

17. n
n
(Limit Comparison)
3
1
n
n
e
(Root)
2

18. n n
)cos(
n n