Fall 2011
Second midterm review sheet
The second midterm covers all the material from chapters 3 – 5, and you
may use results from chapters 1 – 5 without proof. Most of the midterm will
consist of problems from this list. You will not be asked to reprove theorems
from the book.
1. Evaluate
lim
n
→∞
n
√
n
!
n
.
2. Let
p >
1. Evaluate
∞
X
n
=1
n

p
∞
X
n
=1
(

1)
n
n

p
.
3. Let
A >
0, and suppose
∞
X
n
=1
a
n
=
A
, and
a
n
>
0 for all
n
∈
N
. Find the
possible values of
∞
X
n
=1
a
2
n
.
4. Let
∑
b
n
be a convergent series of real numbers, and let (
a
n
)
n
∈
N
be
bounded below. Prove that if
a
n
+1
≤
a
n
+
b
n
,
then (
a
n
)
n
∈
N
converges.
5. Problem 18 from page 100.
6. Problem 25 from page 102.
7. Let
X
and
Y
be metric spaces and
f
a function from
X
to
Y
. Prove
that the following are equivalent.
(a)
f
is continuous.
(b) If (
p
n
)
n
∈
N
is a convergent sequence in
X
, then the sequence (
f
(
p
n
))
n
∈
N
is convergent in
Y
.
(c) If (
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 Fall '10
 Prof.KatrinWehrheim
 Topology, Metric space, Topological space, lim Pn, 2011 SECOND

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