1      2       4        8          16         
   S = --- + ----- + ----- + ------- + ---------- + ..... 
        2      5       41     3,281    21,523,361              

I get the following:

  S_n = 1 - (1/Q_n)

where

  Q_n = 2                           if n = 0

      = Q_(n-1)*(2^n*Q_(n-1) + 1)   if n > 0

I'm still working on a closed form.