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

As others have noted, the numerator is 2^(n-1).

Here's an observation about the denominator.  Let the denominator be D(n) and the numerator be N(n). 

D(n) = (D(n-1) * N(n-1) * D(n-2) * N(n-2) * D(n-3)) + 1

For n=4,

D(4) = (D(3) * N(3) * D(2) * N(2) * D(1)) + 1

D(4) = (41 * 4 * 5 * 2 * 2) + 1 = 3,281

Same works for n = 5:

D(5) = (D(4) * N(4) * D(3) * N(3) * D(2) ) + 1

D(4) = (3,281 * 8 * 41 * 4 * 5) + 1 = 21,523,361

Unfortunately it fails for n=1, 2, and 3.

But it's a start.