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 natkoza December 6th, 2010 07:25 AM

Proof of recurrence sequence convergence

Let $(x_n)_{n\in\mathbb{N}}$ be a recurrence sequecnce such that:

(1) $x_0$ and $x_1$ are positive reals

(2) For $n\geq 2$ we have $x_n=\frac{2}{x_{n-1}+x_{n-2}}$.

Proof that $(x_n)_{n\in\mathbb{N}}$ is convergent.

 CRGreathouse December 6th, 2010 08:51 AM

Re: Proof of recurrence sequence convergence

What happens when the sum is > 1? What happens when the sum is < 1?

 natkoza December 6th, 2010 01:20 PM

Re: Proof of recurrence sequence convergence

From such cases (if you are talking about denominator) follows estimation for the next element of our sequence and I dont see anything more.

I tried to divide it into two subsequences such that 1st is composed of elements greater then one, and 2nd consists of numbers lesser then one. After that if 1st was nonincreasingand 2nd nondecreasing (actually it requiers their convergance only) with some easy calculus I can show that they have a common limit.

My prove attempt distinguished eight cases (there cant be three elements in a row that belong into on of these subsequances) but some of them are troublesome or simply false (for example they may be monotonic only for enought big values).

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