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x
y
For any real number
x
such that
, provethat, for
these
values of
n
,
a
x
x
/
y
, and deduce from the first comparison
test (qn 26) that the series
is absolutely convergent.
It is illuminating to illustrate this result on the number line. If a
power series is convergent for a particular value of the variable
other than 0, then it is absolutely convergent for any value of the
variable which is nearer to the origin.
a
x
y
x
0
x
y
92 The result of qn 91 is available under weaker conditions. Suppose
only that the terms of the series
a
y
arebounded, say,
a
y
K
. Use the argument of qn 91 to prove that the series
a
x
is absolutely convergent when
x
y
.
93 Usethecontrapositiveof qn 91 to show that if
a
y
is divergent,
then
a
x
is divergent when
x
y
.
a
x
94 For a given power series
, we consider the values of
x
for
which the power series is convergent. Every power series is
convergent for
x
0, so the set of such values is not empty.
Let
C
x
:
a
x
is convergent
.
Thest
C
is a set of real numbers
0. Useqn 91 to show that if
C
is unbounded then
a
x
is convergent for all values of
x
.
If
C
is bounded, let
R
denote its least upper bound,
R
sup
C
.
For any
x
with
x
R
, use the properties of a least upper bound
(qn 4.64) to say why there exists a
y
with
x
y
R
and
a
y
a
x
convergent, and deduce that
is absolutely convergent.
If, for some
x
with
x
R
, the series
a
x
were to be convergent,
say why this would contradict thedefinition of
R
as an upper
bound for
C
.
Thework of qn 94 givs thebasis for thedefinition of the
radius of
convergence
,
R
, of a power series
a
x
.
If the power series is only convergent when
x
0, wesay that its
radius of convergence
R
0.
If the power series is convergent for
x
R
and divergent for
x
R
, for some positive real number
R
, then its radius of
convergence is that real number
R
.
If the power series is convergent for all values of
x
, wesay that its
radius of convergence
R
.
When these results are extended to the plane of complex numbers
