Math 1B
First Exam
Name:_________________________
SID:_____________________
(last, first)
1.
For each of the series below, determine whether it converges or diverges.
Use each of the following tests
exactly once:
n
th term test, integral test, direct comparison test, limit comparison test, alternating series test,
ratio test, and root test. Please indicate which test you used on which part.
(10 points for each part)
a)
2
0
n
n
e
∞
−
=
∑
Converges by Direct Comparison Test.
2
2
1
1
n
n
e
<
b)
1
ln
n
n
n
∞
=
∑
Diverges by the Integral test.
(
)
2
1
2
1
1
ln
ln

x
dx
x
x
∞
∞
=
= ∞
∫
c)
(
)
2
0
1
n
n
n
e
∞
−
=
−
∑
Converges by the Alternating Series Test.
d)
(
)
1
2
0
n
n
n
e
∞
−
=
−
∑
Converges by the Root Test.
(
)
(
)
1
1
1
2
2
2
lim
lim
1
n
n
n
n
n
n
e
e
−
−
→∞
→∞
−
=
−
=
<
e)
1
n
n
a
∞
=
∑
where
2
1
1
1
5
4
2,
k
n
n
k
a
a
a
+
+
−
=
=
.
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