CHAPTER 15
Power Series, Taylor Series
Power series and, in particular, Taylor series, play a much more fundamental role in
complex analysis than they do in calculus. The student may do well to review what has
been presented about power series in calculus but should become aware that many new
ideas appear in complex, mainly owing to the use of complex integration.
SECTION 15.1. Sequences, Series, Convergence Tests, page 664
Purpose.
The beginnings on sequences and series in complex is similar to that in calculus
(differences between real and complex appear only later). Hence this section can almost be
regarded as a review from calculus plus a presentation of convergence tests for later use.
Main Content, Important Concepts
Sequences, series, convergence, divergence
Comparison test (Theorem 5)
Ratio test (Theorem 8)
Root test (Theorem 10)
SOLUTIONS TO PROBLEM SET 15.1, page 672
2.
Bounded, divergent, 8 limit points (the values of
Ï
8
1
w
)
4.
Unbounded, divergent
6.
Bounded, convergent to 0 (the terms of the Maclaurin series of
e
3
1
4
i
)
8.
Divergent. All terms have absolute value 1.
10.
Convergent to 0
12.
Let
<
1
and
<
2
be two limits,
d
5
u
<
1
2
<
2
u
and
e
5
d
/3. Then there is an
N
(
) such
that
u
z
n
2
<
1
u
,
,
u
z
n
2
<
2
u
,
for all
n
.
N
.
This is impossible because the disks
u
z
2
<
1
u
,
and
u
z
2
<
2
u
,
are disjoint.
14.
The sequences are bounded,
u
z
n
u
,
K
,
u
z
n
*
u
,
K
. Since they converge, for an
.
0
there is an
N
such that
u
z
n
2
<
u
,
/(3
K
),
u
z
n
*
2
<
*
u
,
/(3
u
<
u
) (
<
Þ
0; the case
<
5
0
is rather trivial), hence
u
z
n
z
n
*
2
<
*
u
5
u
(
z
n
2
<
)
z
n
*
1
(
z
n
*
2
<
*)
<
u
%
u
z
n
2
<
uu
z
n
*
u
1
u
z
n
*
2
<
*
<
u
,
/3
1
/3
,
(
n
.
N
).
16.
Convergent. Sum
e
10
2
15
i
18.
Convergent because
,
and
o
`
n
5
1
converges.
20.
Divergent because 1/ln
n
.
1/
n
and the harmonic series diverges.
22.
By the ratio test it converges because after simplification
jj
5
*
.
Ï
2
w
±
27
(
n
1
1)
3
u
1
1
i
u
±±±
(3
n
1
3)(3
n
1
2)(3
n
1
1)
z
n
1
1
±
z
n
1
±
n
2
1
±
n
2
1
±
u
n
2
2
2
i
u
260
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12:56 PM
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