MATH 104, SUMMER 2006, HOMEWORK 8 SOLUTION
BENJAMIN JOHNSON
Due July 31
Assignment:
Section 23: 23.1(d), 23.7, 23.8
Section 24: 24.1, 24.9, 24.17
Section 23
23.1 Find the radius of convergence and determine the exact interval of convergence.
(d)
∑
n
3
3
n
x
n
Let
a
n
=
n
3
3
n
. Then
a
n
+
1
/
a
n
=
(
n
+
1)
3
3
n
+
1
/
n
3
3
n
=
(
n
+
1)
3
·
3
n
3
n
+
1
·
n
3
=
1
3
·
n
+
1
n
3
From limit laws, we have lim
n
→∞

a
n
+
1
a
n

=
1
3
. So the radius of convergence,
R
, is 3.
to determine whether the interval of convergence is open closed, or halfopen, we need to
consider what happens to the series at
x
=

3 and
x
=
3. For
x
=

3, we have
∑
n
3
3
n
(

3)
n
=
∑
(

1)
n
n
3
which diverges. Similarly,
∑
n
3
3
n
(3)
n
=
∑
n
3
which also diverges. So the interval
of convergence is (

3
,
3).
23.7 For each
n
∈
N
, let
f
n
(
x
)
=
(cos
x
)
n
. Each
f
n
is a continuous function. Nevertheless, show that
(a) lim
f
n
(
x
)
=
0 unless
x
is a multiple of
π
,
Proof.
If
x
is not a multiple of
π
, then

cosx

<
1. So by one of the limit theorems, lim
n
→∞
(cos
x
)
n
=
0.
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 Spring '09
 Math, Calculus, Topology, Limit of a function, FN

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