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7.2 Numerical Series and Sequences
107
Fix
ε >
0. Since
∑
b
n
converges, choose
N
such that
n
≥
N
=
⇒ 
β
n

< ε,
so that

e
n
 ≤ 
a
0
β
n
+
...a
n

N

1
β
N
+1

+

β
N
a
n

N
+
···
+
a
n
β
0

≤
εα
+

β
N
a
n

N
+
···
+
a
n
β
0

.
Now since
a
n
→
0, for
N
ﬁxed and
n >>
1, we can make

β
N
a
n

N
+
···
+
a
n
β
0

< ε
. Then

e
n

<
(
α
+ 1)
ε
.
§
7.2 Exercise: #
Recommended: #
1.
∑
a
n
converges =
⇒
a
n
→
0.
2. If
{
x
n
}
is Cauchy in
R
and some subsequence
{
x
n
k
}
converges to
x
∈
R
, then prove
the full sequence
{
x
n
}
also converges to
x
.
3. If
ﬂ
ﬂ
ﬂ
a
n
+1
a
n
ﬂ
ﬂ
ﬂ
< r
for
n >>
1, then
n
p

a
n

< r
for
n >>
1.
4. If lim

a
n


b
n

= 1, then
X

a
n

converges
⇐⇒
X

a
n

converges
.
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View Full Document 108
Math 413
Sequences and Series of Functions
7.3 Uniform convergence
What does it mean to say a sequence of functions
{
f
n
}
converges? I.e., how to deﬁne
when lim
f
n
(
x
) =
f
(
x
)? There are diﬀerent (nonequivalent) ways to deﬁne such a limit.
What does it mean to say a sum of functions
{
f
n
}
converges? I.e., how to deﬁne
∑
f
n
(
x
) =
f
(
x
)? For example, power series have
f
n
(
x
) =
a
n
x
n
. What about other kinds
of functions?
We want to know when things are valid, like for
f
(
x
) =
∑
f
n
(
x
),
f
0
(
x
)? =
X
f
0
n
(
x
)
Z
f
(
x
)
dx
? =
X
Z
f
n
(
x
)
dx,
The Gamma function is deﬁned Γ(
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.
 Fall '11
 Wong

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