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From: JEMebius on 17 May 2010 09:27
Jos� Carlos Santos wrote:
> On 16-05-2010 12:57, deltasigma wrote:
>
>> suppose that we have a sequence of rational approximations a_n of an
>> irrational number r with a_n<a_(n+1) such that:
>>
>> a_n<a_(n+1)<a_(n+2)<...<r
>>
>> What is the lim(n->oo){a_n} a rational or an irrational number?
>
> It depends. If all that you are saying is that the sequence (a_n)_n is
> increasing and that each of its terms is smaller than _r_, then its
> limit (which must exist), can be any number smaller than or equal
> to _r_.
>
> On the other hand, if the word "approximations" means here that
> lim_n|a_n - r| = 0, then, of course, the limit of the sequence is _r_,
> but then you do not have to assume that the sequence is increasing or
> that each of its terms is smaller than or equal to _r_.
>
> Best regards,
>
> Jose Carlos Santos


Let the irrational number A be the limit of a sequence of rational numbers An = Pn/Qn with
integral Pn, Qn, as proposed by mr Jos� Carlos Santos. Then Qn and therefore Pn too

=will increase in absolute value without bounds=

as one advances along the sequence. This is what happens.
This is virtually equivalent to the irrationality property.

Continued fractions as approximants to irrational numbers provide optimal approximations
and fastest convergence in the sense that in such cases one has

|A - Pn/Qn| < or = 1/(Qn^2 . sqrt(5)).

The worst-behaved irrationals in this respect are the golden ratio sqrt(5/4) - 1/2 and all
other numbers p + q.sqrt(5) with rational p, q.

See for instance http://en.wikipedia.org/wiki/Continued_fraction , in particular the
paragraph on the golden ratio:
http://en.wikipedia.org/wiki/Continued_fraction#A_property_of_the_golden_ratio_.CF.86


Happy studies: Johan E. Mebius
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