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

[Paul D. Hanna]

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