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July 1st, 2012, 07:47 PM   #1
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Convergence of a recurrence relation

Given . Use a calculator to determine the behaviour of the sequence for each of the cases

I had no problems doing and .
For , the series increases and converges to 0.619 while for , the series decreases and converges to 0.619.

However, when solving for , all i got was error on the calculator screen.
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July 1st, 2012, 08:25 PM   #2
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Re: Convergence of a recurrence relation

I observed the same behavior you did, and for the series diverged.

For it increased and converged to 0.619061286736

For it decreased and converged to 0.619061286736

I tried finding the closed form via Newton's method to create a first order IVP, but wound up with an integral without an anti-derivative expressible in elementary terms.

I found by setting that the value the series converged to in the first two cases is one of the solutions to:



The other solution is about 1.5121345516578424739.

Here is a plot:

[attachment=0:5ucqk92t]convergence.jpg[/attachment:5ucqk92t]

The larger root seems to be the division between convergence and divergence for the given recurrence relation when used as the starting value.
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File Type: jpg convergence.jpg (12.2 KB, 360 views)
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July 1st, 2012, 09:02 PM   #3
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Re: Convergence of a recurrence relation

Solving for fixed points,

where W(z) is the Lambert W function, which can be assigned two values there, 0.619061 and 1.51213.
The first is stable and the second is unstable; any starting value greater than 1.51213 will result in divergence.
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July 1st, 2012, 09:10 PM   #4
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Re: Convergence of a recurrence relation

In fairness to [color=#0040FF]aswoods[/color], I was editing my post to include essentially the same information (albeit not as succinctly) in his post at the same time he was posting.
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