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Homework 3 due Thursday October 28 before 13:30 in corre
sponding group folder on door of oﬃce CM110
•
Consider the sequence
{
p
n
}
generated by
p
0
= 0
,
p
n
+1
=

(
p
3
n
+ 2)
/
6
,
n
= 0
,
1
,
2
, ...
Show that for all
n
,

1
3
≤
p
n
≤
0 and that the sequence converges to a limit. Give
a bound for its rate of convergence.
Let
p
n
+1
=
g
(
p
n
) :=

(
p
3
n
+ 2)
/
6. We plan to apply the
convergence theorem
on the
interval [

1
3
,
0]. Notice that
g
0
(
x
) =

x
2
2
≤
0 for
x
∈
[

1
3
,
0]
.
Hence, we conclude that max
x
∈
[

1
3
,
0]

g
0
(
x
)
 ≤
1
18
<
1 and that as
g
is monotone decreasing
so that
g
: [

1
3
,
0]
→
[
g
(0)
, g
(

1
3
)] = [

1
3
,

53
162
]
⊂
[

1
3
,
0]
,
i.e.
g
maps the interval [

1
3
,
0] into itself. Hence we can apply the convergence
theorem which shows that

1
3
≤
p
n
≤
0 for all
n
and that the sequence converges to
a unique limit,
p
. As for a bound on its rate of convergence, since the limit
p
6
= 0,
convergence is linear, since
g
0
(
p
)
6
= 0, and so
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 Fall '11
 Dr.A.Wacher
 Numerical Analysis

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