Riemann zeta and Hypergeometric series relation

Do[
 Print[{N[{Sum[
       Binomial[2 (n - 1), (n - 1)]/4^(n - 1)/n^k, {n, 1, Infinity}] -
       HypergeometricPFQ[Flatten[{{1/2}, Table[1, {n, 1, k}]}],
       Table[2, {n, 1, k}], 1]}]
   }], {k, 1, 12}]