Chapter 7
Power Series
It is a pain to think about convergence but sometimes you really have to.
Sinai Robins
7.1
Sequences and Completeness
As in the real case (and there will be no surprises in this chapter of the nature ‘real versus complex’),
a
(complex) sequence
is a function from the positive (sometimes the nonnegative) integers to the
complex numbers. Its values are usually denoted by
a
n
(as opposed to, say,
a
(
n
)) and we commonly
denote the sequence by (
a
n
)
∞
n
=1
, (
a
n
)
n
≥
1
, or simply (
a
n
). The notion of convergence of a sequence
is based on the following sibling of Deﬁnition 2.1.
Deﬁnition 7.1.
Suppose (
a
n
) is a sequence and
a
∈
C
such that for all
± >
0, there is an integer
N
such that for all
n
≥
N
, we have

a
n

a

< ±
. Then the sequence (
a
n
) is
convergent
and
a
is its
limit
, in symbols
lim
n
→∞
a
n
=
a.
If no such
a
exists then the sequence (
a
n
) is
divergent
.
Example 7.2.
lim
n
→∞
i
n
n
= 0: Given
± >
0, choose
N >
1
/±
. Then for any
n
≥
N
,
±
±
±
±
i
n
n

0
±
±
±
±
=
±
±
±
±
i
n
n
±
±
±
±
=

i

n
n
=
1
n
≤
1
N
< ±.
Example 7.3.
The sequence (
a
n
=
i
n
) diverges: Given
a
∈
C
, choose
±
= 1
/
2. We consider two
cases: If Re
a
≥
0, then for any
N
, choose
n
≥
N
such that
a
n
=

1. (This is always possible since
a
4
k
+2
=
i
4
k
+2
=

1 for any
k
≥
0.) Then

a

a
n

=

a
+ 1
 ≥
1
>
1
2
.
If Re
a <
0, then for any
N
, choose
n
≥
N
such that
a
n
= 1. (This is always possible since
a
4
k
=
i
4
k
= 1 for any
k >
0.) Then

a

a
n

=

a

1
 ≥
1
>
1
2
.
68
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View Full DocumentCHAPTER 7. POWER SERIES
69
The following limit laws are the relatives of the identities stated in Lemma 2.4.
Lemma 7.4.
Let
(
a
n
)
and
(
b
n
)
be convergent sequences and
c
∈
C
.
(a) lim
n
→∞
a
n
+
c
lim
n
→∞
b
n
= lim
n
→∞
(
a
n
+
cb
n
)
.
(b) lim
n
→∞
a
n
·
lim
n
→∞
b
n
= lim
n
→∞
(
a
n
·
b
n
)
.
(c)
lim
n
→∞
a
n
lim
n
→∞
b
n
= lim
n
→∞
±
a
n
b
n
²
.
In the quotient law we have to make sure we do not divide by zero. Moreover, if
f
is continuous at
a
then
lim
n
→∞
f
(
a
n
) =
f
(
a
)
if
lim
n
→∞
a
n
=
a,
where we require that
a
n
be in the domain of
f
.
The most important property of the real number system is that we can, in many cases, determine
that a sequence converges
without knowing the value of the limit
. In this sense we can use the
sequence to deﬁne a real number.
Deﬁnition 7.5.
A
Cauchy sequence
is a sequence (
a
n
) such that
lim
n
→∞

a
n
+1

a
n

= 0
.
We say a
metric space
X
(which for us means
Z
,
Q
,
R
, or
C
) is
complete
if, for any Cauchy sequence
(
a
n
) in
X
, there is some
a
∈
X
such that lim
n
→∞
a
n
=
a
.
In other words, completeness means Cauchy sequences are guaranteed to converge. For example,
the rational numbers are not complete: we can take a Cauchy sequence of rational numbers getting
arbitrarily close to
√
2, which is not a rational number. However, each of
Z
,
R
, and
C
is complete.
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 Spring '10
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 Power Series, Mathematical Series, lim F

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