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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
.
 
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