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

[Gerald McGarvey]

Seqfans,

(*) Pi/2 = [1; 1, 1/2, 1/3, 1/4,..., 1/n,...]
gives the same denominators and numerators as Wallis's approximation to 
Pi/2, see
http://www.research.att.com/~njas/sequences/A001901
http://www.research.att.com/~njas/sequences/A001902
http://mathworld.wolfram.com/WallisFormula.html

Regarding (**), it looks like every other denominator (1,4,36,144,...) of the
convergents for this CF is a denominator of a partial sum of 
Sum_{n=1..infinity} 1/n^2
and that the ratios of the corresponding numerators approaches 1.
numerators:
1, 2, 3, 7, 19, 61, 29, 241, 1169, 5989, 5869, 5969, 41183, 291881, 288731, 
1165949, 128461, 10483741, 10413181, 2095337, 22921699, 253408769, 
252244529, 253311749
denominators:
1, 1, 2, 4, 12, 36, 18, 144, 720, 3600, 3600, 3600, 25200, 176400, 176400, 
705600, 78400, 6350400, 6350400, 1270080, 13970880, 153679680, 153679680, 
153679680

-- Gerald

At 04:18 AM 8/10/2006, Paul D. Hanna wrote:
>Seqfans,
>       Thanks, Joseph, for your comments.
>Note that
>(*) Pi/2 = [1; 1, 1/2, 1/3, 1/4,..., 1/n,...]
>is equivalent to
>Pi/2 = 1 + 1/(1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 +... ))))).
>
>Now it seems like I've seen this expression before ...
>Can anyone provide a reference?
>
>Notice the similarity between (*) and the following:
>
>(**) Pi^2/6 = [1; 1, 1, 1/2, 1/2, 1/3, 1/3, 1/4,..., 1/n, 1/n,...]
>
>  = 1 + 1/(1 + 1*1/(1 + 1*2/(1 + 2*2/(1 + 2*3/(1 + 3*3/(1 +... ))))).
>
>This (**) is conjecture, but is also most likely true and well-known.
>I wonder if the convergents of (**) are as interesting ...
>
>Would appreciate any references that confirm (*) or (**).
>Thanks,
>      Paul