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 Post subject: Contraction sequencePosted: Mon, 20 Feb 2012 06:31:38 UTC
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Joined: Sun, 12 Feb 2012 02:16:54 UTC
Posts: 64
I am suppose to be using the contraction principle to prove that a sequence converges. I am trying to find the a_n+2 term so I can do

|an+2-an+1|<=k|an+1-an|

I am given a1 and an+1, how do I figure out an+2?

My sequence is 1+1/an = an+1

Would my an+2 = 1+1/(an+1) ?

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 06:34:36 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
doodlenoodle wrote:
I am suppose to be using the contraction principle to prove that a sequence converges. I am trying to find the a_n+2 term so I can do

|an+2-an+1|<=k|an+1-an|

I am given a1 and an+1, how do I figure out an+2?

My sequence is 1+1/an = an+1

Would my an+2 = 1+1/(an+1) ?

By definitions , so iterate:

, then look at how the pattern evolves as and you should be able to see how this all terminates.

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 06:56:11 UTC
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Joined: Sun, 12 Feb 2012 02:16:54 UTC
Posts: 64
For using the contraction principle to prove a_n converges tho, I just get something like

|an+2-an+1| = |(1+1/an+1) - (1+1/an)|= |1/an+1-1/an| <= k|(1+1/an - an|

How can I prove that k is between 0 and 1?

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 12:35:32 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
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doodlenoodle wrote:
For using the contraction principle to prove a_n converges tho, I just get something like

|an+2-an+1| = |(1+1/an+1) - (1+1/an)|= |1/an+1-1/an| <= k|(1+1/an - an|

How can I prove that k is between 0 and 1?

You can't, unless you put some restriction on .

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 16:40:03 UTC
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Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 992
doodlenoodle wrote:
For using the contraction principle to prove a_n converges tho, I just get something like

|an+2-an+1| = |(1+1/an+1) - (1+1/an)|= |1/an+1-1/an| <= k|(1+1/an - an|

How can I prove that k is between 0 and 1?

Would showing that the sequence is Cauchy be enough to show it satisfies the Contraction Principle?
This sequence clearly converges.

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 16:59:17 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12103
Location: Austin, TX
Justin wrote:
doodlenoodle wrote:
For using the contraction principle to prove a_n converges tho, I just get something like

|an+2-an+1| = |(1+1/an+1) - (1+1/an)|= |1/an+1-1/an| <= k|(1+1/an - an|

How can I prove that k is between 0 and 1?

Would showing that the sequence is Cauchy be enough to show it satisfies the Contraction Principle?
This sequence clearly converges.

Why should it?

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 Post subject: Re: Contraction sequencePosted: Mon, 20 Feb 2012 17:44:11 UTC
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Joined: Wed, 21 May 2003 04:27:18 UTC
Posts: 992
Justin wrote:
doodlenoodle wrote:
For using the contraction principle to prove a_n converges tho, I just get something like

|an+2-an+1| = |(1+1/an+1) - (1+1/an)|= |1/an+1-1/an| <= k|(1+1/an - an|

How can I prove that k is between 0 and 1?

Would showing that the sequence is Cauchy be enough to show it satisfies the Contraction Principle?
This sequence clearly converges.

Why should it?

Just a thought...although I see your point. If showing the sequence Cauchy was that easy, you wouldn't need to use the
Contraction Principle to begin with. I'm guessing a triangle inequality estimate is required here.

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

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