Sequences is defined with

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- Jan 21st 2010, 10:17 AMns1954Limit 1
Sequences is defined with

Find - Jan 21st 2010, 12:13 PMchisigma
The possible finite limits of the sequence are the solutions of the equation...

(1)

... that are...

(2)

A quickly inspection reveals that if is...

(3)

Kind regards

- Jan 21st 2010, 02:11 PMns1954
Yes, that is limit. But how to prove convergence of this, using therem of monoton and bounded sequences? It's not quite easy, it must be considered, I think, subsequences

and

One is increasing, second decreasing, and both oscilate about . This may be a strict proof. - Jan 21st 2010, 09:46 PMchisigma
It is not at all difficult to prove that...

if or

if

if (1)

On the basis of these unequalities we have the following distinct cases...

a) if the sequence diverges at

b) if is ,

c) if the sequence converges at in any case with 'oscillatory behavior' unless is that produces the constant sequence , ...

Kind regards

- Jan 22nd 2010, 05:04 AMns1954

,

,

,

,

,

,

,

,

,

and you can "see " that , and "is" monotonicaly increasing, and , and "is" monotonicaly decreasing.

So, here is neccessary only strict proof of this monotonity. Maybe this relation

Observe that , (by inicial value) for all , so second parenteze is negative, and somehow here must be used induction.

I belive similar can be done for

and

Update: Hear is another relation

, so this could be used maybe (I think) for variant of induction (with assumptation for and , and proff for and ).