Approximation
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
3.05 Barriers to Convergence
LITERACY
·
L.1)
At what point on the xaxis are the convergence intervals of expansions in powers of
H
x

1
L
centered?
At what point on the xaxis are the convergence intervals of expansions in powers of
H
x
+
5
L
centered?
·
L.2)
Use complex numbers and singularities to explain why, when you expand
f
@
x
D
=
1
I
1
+
x
2
M
in powers of x, you run into barriers at x
= 
1 and x
=
1.
·
L.3)
Use complex numbers and singularities to explain why, when you expand
f
@
x
D
=
ArcTan
@
x
D
in powers of x, you run into barriers at x
= 
1 and x
=
1.
Confused? Look at the tip below:
D
@
ArcTan
@
x
D
, x
D
1
1
+
x
2
·
L.4)
Use complex numbers and singularities to explain why, when you try to expand
f
@
x
D
=
x
H
5
ê
2
L
in powers of x, you must fail.
·
L.5)
Give the convergence interval for the expansions of the following functions
f
@
x
D
in powers
of
H
x

a
L
for the given choices of
a
:
Ø
f
@
x
D
=
1
1

x
with
a
=
0
,
Ø
f
@
x
D
=
1
1

x
with
a
=
4
,
Ø
f
@
x
D
=
e
x
with
a
=
0
,
Ø
f
@
x
D
=
Sin
@
x
D
with
a
=
0
,
Ø
f
@
x
D
=
Sin
@
x
D
with
a
= p
,
Ø
f
@
x
D
=
Cos
@
x
D
with
a
=
0
,
Ø
f
@
x
D
=
1
+
x
with
a
=
0
,
Ø
f
@
x
D
=
x
I
5
3
M
with
a
=
1
,
Ø
f
@
x
D
=
ArcTan
@
x
D
with
a
=
0
, and
Ø
f
@
x
D
=
1
1
+
x
2
with
a
=
1
.
·
L.6)
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 Spring '08
 Kim
 Approximation, Convergence, Complex number

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