Generalized Harmonic numbers perturbed by an epsilon divided by 2 to the power s

(* start *)
(* Mathematica 8.0.1 *)
n = 20;(*set n=30 for better precision*)
epsilon = 1/10^10;
s = 1 + 14*
   I;(*change the number s=1+14*I to get another zero closest to it*)
\

s + 1/n +
 1/(1 - Sum[(-1)^(k - 1)*
       Binomial[n - 1,
         k - 1]/(HarmonicNumber[10^10000, s + k/n] +
          epsilon/2^(s + k/n)), {k, 1, n}]/
     Sum[(-1)^(k - 1)*
       Binomial[n - 1,
         k - 1]/(HarmonicNumber[10^10000, s + k/n + 1/n] +
          epsilon/2^(s + k/n + 1/n)), {k, 1, n}]);
N[%, 15]
(* end *)