Graphics Reference
In-Depth Information
a
x
r
1/
A
, for suMciently large
n
,
1/
r
, so wehave
a
x
x
/
r
1, and so the series
a
x
is convergent. Identify
the radius of convergence.
The exploration in questions 99, 100 and 102 leads to the
Cauchy—Hadamard formula
R
1/lim sup
a
,
which, with suitable interpretations, is a general statement of the radius
of convergence of the power series
a
x
.
103 Give an example of a power series
a
x
, with radius of
convergence 1, which is divergent at each point on the circle of
convergence.
104 Give an example of a power series
a
x
, with radius of
convergence 1, which is convergent at one point on the circle of
convergence, and divergent at another.
105 Give an example of a power series
a
x
, with radius of
convergence 1, which is convergent at each point on the circle of
convergence.
106 Compare the regions of convergence of the three power series
x
2
x
3
x
4
...,
log(1
x
)
x
1
1
x
1
x
x
x
...,
1
1
2
x
3
x
4
x
....
(1
x
)
107 UsetheCauchy
—
Hadamard formula to show that the three series
a
x
/(
n
1) havethesameradius of
convergence. This establishes that the radius of convergence is not
changed when a power series is differentiated or integrated term by
term.
x
,
na
x
and
a
The Cauchy product
108 If
A
(
x
)
a
x
and
B
(
x
)
b
x
, calculatethe
coeMcients,
c
, in theproduct
x
x
A
(
x
) ·
B
(
x
)
c
d
.
