f[n_] := (n + 2)/2^(n + 2)*
  Sum[Binomial[n - k + 1, k], {k, 0, Floor[n/2]}]
Table[
 Sum[
   f[n],
   {n, 0, max}
   ] // N,
 {max, 0, 200, 10}
 ]

Result:
0.5, 6.32422, 8.40171, 8.80889, 8.8767, 8.88711, 8.88864, 8.88885, \
8.88888, 8.88889, 8.88889, 8.88889, 8.88889, 8.88889, 8.88889, \
8.88889, 8.88889, 8.88889, 8.88889, 8.88889, 8.88889