A092053: Value of Continued Fraction [1;1/2,1/3,1/4,...,1/n,...]

[Paul D. Hanna]

Seqfans, 
      My guess is: 
 
(*) [1;1/2,1/3,1/4,...,1/n,...] = 1/(Pi/2 - 1) = 1.7519383938841...
 
which would explain the behavior of the numerators and 
denominators of the convergents. 
 
Proof of (*) anyone? 
 
On Wed, 9 Aug 2006 19:53:56 -0400 "Paul D. Hanna" <pauldhanna at juno.com>
writes:
> Seqfans, 
> Can anyone evaluate the continued fraction:
>    x = [1;1/2,1/3,1/4,...,1/n,...].
>  
> Convergence is very slow: 
> at 500000 partial quotients, x = 1.751943215111853159301... ;
> at 600000 partial quotients, x = 1.751942411573727042118...
>  
> The convergents of the CF are interesting, and are related to Pi/2.
>  
> See A092053: 
> "Denominators of the convergents of the continued fraction 
> expansion
> [1;1/2,1/3,1/4,...,1/n,...]."
> COMMENT: 
> "Numerators of convergents are A001902 (successive denominators 
> of Wallis's product approximation to Pi/2). 
> Sum of numerators and denominators equals powers of 2: 
> A001902(n) + a(n) = 2^A092054(n)."
>  
> Thanks, 
>    Paul