Math 231 Practice Exam 2 SOLUTIONS
Problem 1:
State as briefly and concisely as you can the following theorems/tests. List
all hypothesis and the conclusions.
Your statement should be formulated “If
(hypothesis) Then (conclusions).” If it is possible for the test to fail so indicate.
(i)
Integral Test
(
Hypothesis
)
Suppose that
f
(
x
)
is a positive, decreasing function and
a
n
=
f
(
n
)
.
then (
Conclusion
) the series
∑
a
n
and the (improper)integral
∞
1
f
(
x
)
dx
either both converge or both diverge.
(ii)
Limit Comparison Test
Hypothesis
Suppose that
a
n
>
0
for all
n
and
lim
a
n
b
n
=
L >
0
Then
(Con
clusion)
the series
∑
a
n
and
∑
b
n
either both converge or they both diverge.
(iii)
Comparison Test for SEQUENCES
Hypothesis
Suppose that
0
< a
n
≤
b
n
for all
n
(sufficiently large) Then
(Conclusions)
•
∑
b
n
converges implies
the series
∑
a
n
converges
•
∑
a
n
diverges implies
the series
∑
b
n
diverges
1
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(iv)
k
th
Term Test
There are two ways to state this: the way the test is
stated in the book or the contrapositive, which is how it is usually used
(
Hypothesis
) Suppose
∑
a
n
converges. then (
Conclusion
)
lim
n
→∞
a
n
=
0
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 Spring '08
 Bronski
 Math, Calculus, Mathematical analysis, 1 K, 3 k

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