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(ii) Provethat
·
d
a
b
.
(iii) Provethat
is absolutely convergent. Use the product rule,
qn 3.54, and also 3.80.
d
(iv) Show that theCauchy product of
a
and
b
is a
rearrangement of
d
.
(v) Provethat theCauchy product of
a
and
b
is convergent
to thesum
A
·
B
.
112 By calculating theCauchy product of
with itself when
1
x
1, provethat 1/(1
x
)
(
n
1)
x
. The series begins
with
n
x
0.
113 Prove the Binomial Theorem for negative integral index by
induction using a Cauchy product, or in other words, give a power
series which converges to 1/(1
x
)
when
1
x
1.
Summary
-
power series and the Cauchy product
Theorem
qns 91, 92
If thetrms of
a
y
are bounded, then
a
x
is
absolutely convergent when
x
y
.
Definition
When
x
:
a
x
is convergent
is bounded
above, sup
x
:
a
x
is convergent
is called
the
radius of convergence
of
a
x
. When
x
:
a
x
is convergent
is not bounded
above, the radius of convergence of
a
x
is
said to beinfinit.
Radius of convergence
qn 94
When
x
the radius of convergence,
a
x
is
absolutely convergent.
The Cauchy—Hadamard formula
qn 102
The radius of convergence of
a
x
is
1/lim sup
a
.
Theorem
qn 107
If a power series is differentiated or integrated
term by term, its radius of convergence does not
change.
Definition
qn 108
If
c
a
b
a
b
...
a
b
...
a
b
,
then the series
c
is called the
Cauchy
product
of the series
a
and
b
.
