Does the sequence defined by 'a_(1)=2,a_(n+1)=(72)/(1+a_(n))' comverge? What does it converge to?The sequence diverges to infinity" There is not enough information to determine the comergence of the sequenceThe sequence comverges to 8The sequence comerges to 10
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Answer:
If the sequence converges then
a
n
+
1
=
a
∞
=
72
1
+
a
n
=
72
1
+
a
∞
or
a
∞
=
72
1
+
a
∞
and solving for
a
∞
we get at
a
∞
=
{
−
9
,
8
}
Now analyzing the behavior of the transformation
f
(
x
)
=
72
1
+
x
we have
f
'
(
x
)
=
−
72
(
1
+
x
)
2
and
|
f
'
(
8
)
|
=
8
9
<
1
⇒
8
is a stable sequence limit point
|
f
'
(
−
9
)
|
=
9
8
>
1
⇒
−
9
is an unstable sequence limit point.
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