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47 with qn 43,
d
'
Alembert
'
s ratio test
Suppose that we have a series of positive terms
a
for which the
sequence of ratios (
a
/
a
)
k
1. By a suitablechoiceof
0
(what
?), show that there exists an
N
such that
n
N
1
.
apply question 46 and the start rule (qn 16) to show that
(1
k
)
a
/
a
a
is
divergent.
48 For which positive real numbers
x
is the series
convergent
and for which is it divergent? Use both d'Alembert's ratio test and
Cauchy's
n
th root test.
(2
x
)
n
It is important to notethat
d'Alembert's ratio test
(qns 43 and 47) gives
no information about the convergence of
a
)
1. So, in
particular, d'Alembert's ratio test gives no information about the
convergence or divergence of
1/
n
,
1/
n
or
1/
n
.
when (
a
/
a
49 Givean exampleto show that thecondition 0
a
/
a
1 is not
suMcient to ensure the convergence of the series
a
.
50 By considering the sequence defined by
a
1/(2
n
)
and
a
1/(2
n
)
, show that a series
a
may be convergent even when
the sequence (
a
/
a
) is unbounded.
Second comparison test
n
1
1
n
51 If
a
and
b
,
n
2
provethat (
a
.
Deduce that for some
N
,
n
N
a
/
b
)
1as
n
/
b
.
Now use the convergence of
b
to establish the convergence of
3/2
b
(with the scalar rule, qn 19) and thus the convergence of
a
.
n
1
1
n
,
52 If
a
and
b
n
2
provethat (
a
)
1as
n
.
Deduce that for some
N
,
n
N
a
/
b
/
b
. Now usethe
divergence of
b
(with the
contrapositive of the scalar rule, qn 19) and deduce the divergence
of
to establish divergence of
b
a
.