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Handout 11 Exam 3 Review Page 1 of 3
Handout 11
MATH 172 Lab: Sections 7 and 8
Lab Instructor (TA): Mohammed Kaabar
Note: This handout is a review for exam 3 in MATH 172.
The following is a summary of convergence and divergence tests for series
Test
Series
Convergent
Divergent
Notes
You can use
to compare
with original
series as we
do in the
comparison
test
Cannot be used for
showing convergence
WARNING:
We say no-
conclusion if
If it is
convergent,
then you need
to write the
sum as:
. You
can also use
this test for
direct and
limit
comparison
tests
If
is
convergent, then
is also
convergent
Cannot be
used for
showing
divergence
If you have a
series that has
a
combination
of positive
and negative
terms, then
this test can
work well
Handout 11 Exam 3 Review Page 2 of 3
Cannot be
used for
showing
divergence
If it is
convergent,
then you need
to write the
sum as:
where is
the initial
term (1st term
in series)
Three Conditions*:
a. Alternating
b. Decreasing
(
).
c.
Cannot be
used for
showing
divergence
The
remainder
can be found
as follows:
where is
continuous,
positive, and
decreasing
is convergent if
and only if
is convergent
is
divergent if
and only if
is divergent
The
remainder
can be found
as follows:
If
is convergent
and for
every , then
is
convergent
If
is
divergent
and
for every
, then
is
divergent
and
is convergent if
is
divergent if
and
Test is
inconclusive
if
Test is
inconclusive if
Handout 11 Exam 3 Review Page 3 of 3
* To determine whether the alternating series is absolutely convergent or conditionally
convergent, you need to use the following Method:
Mohammed Kaabar Binary Method for Alternating Series Test:
Type
1 Convergent
1 Convergent
Absolutely Convergent
1 Convergent
0 Divergent
Conditionally Convergent
0 Divergent
0 Divergent
Divergent
0 Divergent
1 Convergent
Inconclusive