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MATH 104, SUMMER 2006, HOMEWORK 5 SOLUTION
BENJAMIN JOHNSON
Due July 19
Assignment:
Section 14: 14.4, 14.7, 14.10, 14.13(d)
Section 15: 15.4, 15.6, 15.7
Section 14
14.4 Determine which of the following series converge. Justify your answer.
(a)
∑
∞
n
=
2
1
[
n
+
(

1)
n
]
2
The series converges.
I’ll use the comparison test. We have
±
±
±
±
1
[
n
+
(

1)
n
]
2
±
±
±
±
≤
1
(
n

1)
2
for every
n
≥
2. Since
∑
∞
n
=
2
1
(
n

1)
2
=
∑
∞
n
=
1
1
n
2
converges, so does
∑
∞
n
=
2
1
[
n
+
(

1)
n
]
2
by the comparison test.
(b)
∑
[
√
n
+
1

√
n
]
The series diverges.
By the telescoping sum formula,
∑
N
j
=
0
[
√
n
+
1

√
n
]
=
√
N
+
1

√
0. So
∑
∞
j
=
0
[
√
n
+
1

√
n
]
=
lim
n
→∞
∑
N
j
=
0
[
√
n
+
1

√
n
]
=
lim
n
→∞
√
N
+
1
=
∞
.
(c)
∑
n
!
n
n
The series converges.
I’ll use the ratio test. We have
±
±
±
±
±
±
±
(
n
+
1)!
(
n
+
1)
n
+
1
n
!
n
n
±
±
±
±
±
±
±
=
²
n
n
+
1
³
n
=
²
n
n
+
1
³
n
+
1
·
n
+
1
n
=
´
1

1
n
+
1
!
n
+
1
·
n
+
1
n
Taking the limit as
n
→ ∞
and applying the limit product law yields
1
e
·
1
=
1
e
<
1. So the
series converges by the ratio test.
14.7 Prove that if
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