Graphics Reference
In-Depth Information
The series
c
is said to bethe
Cauchy product
of the series
a
and
b
.
109 By putting
a
b
(
1)
/
(
n
1), show that even if the series
a
and
b
are convergent, their Cauchy product need not be
convergent.
110 EvaluatetheCauchy product of thesris
a
and the series
b
when
a
x
/
n
! and
b
y
/
n
!, showing that
c
(
x
y
)
/
n
!.
With the result of qn 111, qn 110 illustrates that
exp(
x
) · exp(
y
)
exp(
x
y
), a result which we will establish by a
different route in chapter 11. See qn 11.20(i) and 11.27
.
a
A
. The series
111 The series
is absolutely convergent and
a
b
B
. Wewill show that
the Cauchy product of such series is convergent to the sum
A
·
B
.
The first array below is an array of products from these two series;
the second array is a renaming of the first array, position by
position.
is absolutely convergent and
b
a
b
a
b
a
b
a
b
...
a
b
...
a
b
a
b
a
b
a
b
...
a
b
...
a
b
a
b
a
b
a
b
...
a
b
...
a
b
a
b
a
b
a
b
...
a
b
...
a
b
a
b
a
b
a
b
...
a
b
...
d
d
d
d
...
d
...
d
d
d
d
...
d
...
d
d
d
d
...
d
...
d
d
d
d
...
d
...
d
d
d
d
...
d
...
(i) Provethat
·
d
a
b
.
