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

[Paul D. Hanna]

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