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 Post subject: Contraction principlePosted: Tue, 20 Mar 2012 22:39:01 UTC
 Senior Member

Joined: Sun, 12 Feb 2012 02:16:54 UTC
Posts: 64
Hello,

Consider the sequence , where and , for all n in N.

a) is monotone?
b) Use the contraction principle to show that converges.
c) Verify that converges to radical 2.

a) No not monotone since a1<a2>a3<a4>a5... and so on.
b) where k is between 0 and 1.

=

for all n greater than 1

Since , then for all n

Thus > (2)(2) = 4
So here we take k=1/4 satisfying the contraction principle.

c) Since converges to some L, we see that

Does this suffice as an answer?

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 Post subject: Re: Contraction principlePosted: Wed, 21 Mar 2012 01:46:24 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
doodlenoodle wrote:
Hello,

Consider the sequence , where and , for all n in N.

a) is monotone?
b) Use the contraction principle to show that converges.
c) Verify that converges to radical 2.

a) No not monotone since a1<a2>a3<a4>a5... and so on.
b) where k is between 0 and 1.

=

for all n greater than 1

Since , then for all n

Thus > (2)(2) = 4
So here we take k=1/4 satisfying the contraction principle.

c) Since converges to some L, we see that

Does this suffice as an answer?

You need to state that L>0, which is easy, but necessary.

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 Post subject: Re: Contraction principlePosted: Wed, 21 Mar 2012 01:53:16 UTC
 Senior Member

Joined: Sun, 12 Feb 2012 02:16:54 UTC
Posts: 64
Otherwise it looks good?

Also, finally getting into the latex format.

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 Post subject: Re: Contraction principlePosted: Wed, 21 Mar 2012 01:55:52 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
doodlenoodle wrote:
Otherwise it looks good?

Also, finally getting into the latex format.

Pro tip: when doing limits, start off your code box with a \$ it changes into

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