Polynomial like Dirichlet eta and Euler Maclaurin

(*Mathematica start*)
x = N[Exp[-ZetaZero[1]/10], 100]
Sum[(-1)^(k - 1)*x^(10*Log[k]), {k, 1, Infinity}]
Limit[Sum[x^(10*Log[n]), {n, 1, k}] - (k x^(10 Log[k]))/(
  1 + 10 Log[x]), k -> Infinity]
(*end*)