Graphics Reference
In-Depth Information
By choosing
for the sequence of ratios, show that the method
of qn 40 can be applied here to prove that any such sequence is
null.
0
l
ε
l
l
+ ε
1
71 Let (
a
) be a sequence of positive terms such that
a
a
l
1.
Show that, for suMciently large
n
, the sequence (
a
) is squeezed
between 0 and an appropriately chosen null geometric progression.
Deduce that (
a
) is a null sequence.
72 Let (
a
) be a sequence of positive terms such that
a
a
l
1.
Show that, for suMciently large
n
, the terms of the sequence (
a
) are
greater than the corresponding terms of an appropriately chosen
geometric progression, which from qn 21 tends to
.
Deduce that (
a
)
.
73 Give an example of a null sequence of positive terms (
a
) for which
a
a
1.
Also give an example of an increasing non-convergent sequence of
positivetrms (
a
) for which
a
a
1.
This means that the condition that the ratio of successive terms
tend to 1 does not determine the convergence of the original
sequence one way or the other.
74 Let
k
be a given integer. Determine for what values of
x
the
sequences with
n
th terms
(ii)
x
(i)
n
x
and
n
!
arenull.
Can thevalueof
k
affect the answer?
