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a
a
37 If
1 for all
n
, provethat
is divergent.
38 A series of positive terms
a
is given, for which the sequence
(
a
)
k
as
n
, where 1
k
. Prove that the series is divergent.
39 By considering the two series
, show that Cauchy's
n
th root test gives no information about the convergence of the
series, when
k
1 in qn 38.
1/
n
and
1/
n
d'Alembert's ratio test
40 Let
a
n
/2
. Provethat if
n
3, then
a
/
a
. By using this
inequality when
n
3, 4, 5, . . . provethat
a
. Usequstion
20 (geometric series) and the first comparison test to show that
a
(
)
a
is convergent. Then use the start rule (qn 16) to prove that
a
is convergent.
41 Let
a
n
(0.8)
. Provethat if
n
14, then
a
/
a
0.99. Use
arguments like those in qn 40 to prove that
a
(0.99)
a
.
a
Deduce that
is convergent.
42 To generalise the results of qns 40 and 41, we suppose that we have
a series
a
of positivetrms for which
0
a
1, for all
n
.
Show that
a
/
a
. Use geometric series (qn 20) and the first
comparison test to prove that
a
·
a
is convergent.
0
k
ε
kk +
ε
1
43
d
'
Alembert
'
s ratio test
, with qn 47
Supposethat
a
is a series of positive terms and the sequence of
ratios (
a
/
a
)
k
1. By a suitablechoiceof
0 (what
?),
show that there exists an
N
such that
n
N
a
(
k
/
a
1)
1.
Let
(
k
1), and apply qn 42 and thestart rule(qn 16) to
provethat
a
is convergent.
44 Provethat
2
/
n
! is convergent.
45 Provethat
n
!/
n
is convergent.
46 Suppose that we have a series of positive terms
a
for which
1
a
/
a
for all
n
. Provethat
a
a
for all
n
. Deduce that (
a
)
is not a null sequence and so
a
is divergent.
