n = 100;
s = 2 + I*14;
(s + 1/n +
1/(1 - Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/HarmonicNumber[10^10000, s + k/n], {k,
1, n}]/Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/
HarmonicNumber[10^10000, s + k/n + 1/n], {k, 1, n}]));
N[%, n/2]
s = 2 + I*14;
(s + 1/n +
1/(1 - Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/HarmonicNumber[10^10000, s + k/n], {k,
1, n}]/Sum[(-1)^(k - 1)*
Binomial[n - 1, k - 1]/
HarmonicNumber[10^10000, s + k/n + 1/n], {k, 1, n}]));
N[%, n/2]