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109 The two series are convergent by the alternating series test.
c
(
n
1)/(
(
n
1))(
(
n
1))
1. Not convergent by the null
sequence test.
110 Use qn 1.5 for the proof. After completing qn 111, use qn 86 to
undrstand thesignificanceof thersult.
111
(i) Usesimplealgebra.
(ii) Usesimplealgebra.
(iii) Thepartial sums of
form an increasing sequence, with a
convergent subsequence from (ii) and theproduct rule3.54(vi).
From 3.80,
d
d
is convergent, and so
d
is absolutely
convergent.
(iv) Cauchy product
d
(
d
d
)
(
d
d
d
)
...
(v) Since
is absolutely convergent, any rearrangement is
absolutely convergent to the same sum by qn 77.
d
112
x
is absolutely convergent when
x
1 from qn 9. In theCauchy
product
c
is a sum of
n
1 terms.
113 Suppose
n
(
n
1)
1 · 2
n
(
n
1)...(
n
r
1)
r
!
(1
x
)
1
nx
x
...
x
....
This series has radius of convergence 1 by qn 98. The coe
M
cient of
x
in theCauchy product with (1
x
)
is
n
(
n
1)
1 · 2
...
n
(
n
1)...(
n
r
1)
r
!
1
n
n
2
n
(
n
2)...(
n
r
1)
(
n
1)
1
...
r
!
(
n
1)(
n
2)
n
3
n
(
n
3)...(
n
r
1)
1
...
2
3 · 4 · ...·
r
(
n
1)(
n
2)...(
n
r
)
,
r
!
which is thesameform as that assumed for
n
. When
n
1, the series
takes the familiar form for a geometric progression. This, by induction,
establishes the expansion for
n
.
