Re: 2.058833711777943011170351... = ? continued fraction convergents//

raymond manzoni a écrit :


Nice! The next one is :
+1/(17*53*109*2269*4373*19441*448607550257*16000411124306403070561) of course!

the denominators are the products of a(n) with
a(n)= a(n-1)^3 + 3a(n-1)^2 - 3
(well 1/1 should be 1/2 but....)

the sum is simply :
1 + 2* sum_{n>=0) 1/Fib(3^n)

http://www.research.att.com/~njas/sequences/A002814

Raymond


So that the inspirating problem was probably the evaluation of

sum_{n>=0} 1/F_{2^n} = (7 - sqrt(5))/2
(Millin “Problem H–237”, The Fibonacci Quarterly 1974)
http://www.mathpropress.com/stan/bibliography/reciprocalGeomProgr.pdf

Note that we have too :
sum_{n=0}^m 1/F_{2^n} = 3 - F_{2^m-1}/F_{2^m}

http://pweb.nju.edu.cn/zwsun/39f.pdf

That should help to find an exact answer for the last one... if this may be done.

Fine search!

Raymond
.

Relevant Pages

  • Re: 2.058833711777943011170351... = ? continued fraction convergents//
    ... raymond manzoni wrote: ... the sum is simply: ... --- Christopher Heckman ...
    (sci.math)
  • Re: Reciprocals of the natural numbers
    ... How many of the denominators are a multiple of this ... What can be inferred when you multiply the whole sum by the common ... I'd say, this proof does not rely on the order property, ...
    (sci.math)
  • Re: Reciprocals of the natural numbers
    ... find the highest power of 2 that is a factor of one of ... How many of the denominators are a multiple of this ... What can be inferred when you multiply the whole sum by the common ...
    (sci.math)
  • Re: Reciprocals of the natural numbers
    ... How many of the denominators are a multiple of this ... What can be inferred when you multiply the whole sum by the common ... Betrand's postulate does give a proof.(the desired p lies ... I want to avoid using the order property. ...
    (sci.math)
  • Re: Reciprocals of the natural numbers
    ... find the highest power of 2 that is a factor of one of ... How many of the denominators are a multiple of this ... What can be inferred when you multiply the whole sum by the common ...
    (sci.math)