UC Davis, Spring 2011
Math 25 — Solutions to Homework Assignment #9
1. Decide whether or not each of the following inﬁnite series converges or diverges. Prove your claims.
(a) The series
∞
∑
n
=1
1
n
√
n
diverges.
Proof.
Consider the sequence
a
n
=
n
√
n
. Note that
lim
n
→∞
log
(
n
√
n
)
= lim
n
→∞
log
n
n
= 0
,
which implies
a
n
→
1. So, by the
n
th term test (also called the trivial test in the textbook), the series
diverges.
(b) The series
∞
∑
n
=1
1
n
n
converges.
Proof.
Note that
n
k
≤
n
j
for
n,k,j
∈
N
and
k < j
, so that
1
n
k
≥
1
n
j
. Then,
∞
X
n
=1
1
n
n
= 1 +
∞
X
n
=2
1
n
n
≤
1 +
∞
X
n
=2
1
n
2
,
so the series converges by the comparison test.
(c) The series
∞
∑
n
=1
n
(
n
+1)
(
n
+2)
2
diverges.
Proof.
Note that
lim
n
→∞
n
(
n
+ 1)
(
n
+ 2)
2
= 1
,
so the series diverges by the
n
th term test.
(d) The series
∞
∑
n
=1
3
n
(
n
+1)(
n
+2)
n
3
√
n
diverges.
Proof.
Note that
∞
X
n
=1
3
n
(
n
+ 1)(
n
+ 2)
n
3
√
n
=
∞
X
n
=1
3
n
3
+ 9
n
2
+ 6
n
n
3
√
n
=
∞
X
n
=1
3
n
3
n
3
√
n
+
9
n
2
n
3
√
n
+
6
n
n
3
√
n
≥
3
∞
X
n
=1
1
√
n
,
which diverges by the
p
series test. So, by the comparison test, the original series diverges.
1
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 Winter '11
 Wiley
 Differential Equations, Calculus, Equations

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