[seqfan] Re: Can π be approximated in this manner?

israel at math.ubc.ca israel at math.ubc.ca
Thu Oct 15 21:09:20 CEST 2020


The sequence b(n)=A073833(n)/A073834(n) with b(n+1) = b(n) + 1/b(n)
has b(n) ~ sqrt(2n) as n -> infinity (as implied by Jon Schoenfield's 
comment of Dec 15 2013).  

To have a sequence c(n) with c(n) = n + o(n), you'll need 
c(n+1) - c(n) = 1 + o(n).

Cheers,
Robert

On Oct 15 2020, Alonso Del Arte wrote:

>Obviously 1 + 1 = 2. Then 2 + 1/2 = 5/2, 5/2 + 2/5 = 29/10, 29/10 + 10/29 =
>941/290, etc. This leads to A073833 / A073834. I don't know if that
>converges to anything in particular or just keeps growing. But the thought
>occurs to me that if sequences of this type do converge, maybe there's one
>that approximates π, or π multiplied by some obvious positive rational
>number.
>
>I tried a few, without much success, e.g.,
>
> scala> res27.take(10).toList res28: List[fractions.BigFraction] = 
> List(1/30, 901/30, 812701/27030, 661213536301/21967308030, 
> 437685903209758747243501/14525081425529653797030, 
> 191779927858960569002522847057449146662223557901/6357423382928236658859400112096701350640602030, 
> ...
>
>scala> res28.map(_.getDouble)
>res29: List[Double] = List(0.03333333333333333, 30.033333333333335,
>30.06662967073622, 30.099889135164094, 30.133111848892693,
>30.166297933523268, 30.199447509987827, 30.23256069855425,
>30.265637618831395, 30.29867838977414)
>
>


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