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Homework 6 – Math CS 120, Winter 2009
Due on Thursday, February 19th, 2009
1. Given
A
⊂
R
n
, and
z
∈
R
n
, prove that the following statements are
equivalent:
(a)
z
∈
A
(the closure of
A
, also denoted by
cl
(
A
)).
(b) There exists a sequence
{
x
n
}
n
∈
N
⊂
A
such that
x
n
→
z
.
(c) For any
± >
0,
A
∩
B
(
z
, ±
)
6
=
{∅}
.
2. Prove that for any
A
⊂
R
n
,
∂A
=
cl
(
A
)
\
int
(
A
), where
cl
(
A
) denotes
the closure of
A
, and
int
(
A
) denotes its interior.
3. Consider a convergent sequence,
{
x
n
} ⊂
R
p
, and its limit,
z
∈
R
p
.
Deﬁne the set
A
=
{
x
n
}
n
∈
R
p
∪ {
z
}
. Prove, using the deﬁnition in
terms of open coverings, that
A
is compact.
4. Show that
2
1

p
<
x
p
+
y
p
(
x
+
y
)
p
<
1
for any
x >
0,
y >
0, and
p >
1.
5. Discuss the convergence behavior of the following series stating in each
case whether the series is absolutely convergent, conditionally conver
gent, or divergent:
(a)
∞
X
n
=1
1
n
!
.
(b)
∞
X
n
=3
n
(
n
+ 1)(
n
+ 2)
(
n
+ 3)
3
.
(c)
∞
X
n
=1
±
1 +
1
n
2
²
n
2
.
(d)
∞
X
n
=1
(

1)
n
√
n
.
1
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(e)
∞
X
n
=1
log
(
1 +
1
n
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 Fall '09
 Math

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