Sum[(-1)^(k + 1)*x^(Log[k]*h), {k, 1, Infinity}]

Clear[k, x, k];
h = 10^100;
Sum[(-1)^(k + 1)*x^(Round[Log[k]*h]), {k, 1, 10}]
x = N[Exp[-(ZetaZero[1])/h], 300]
(*ListLinePlot[Table[Re[(-1)^k*x^(Log[k]*h)],{k,1,1000}],PlotStyle->\
Pink]*)
(*ListLinePlot[Table[Re[(-1)^k*x^(Round[Log[k]*h])],{k,1,1000}]]*)
Sum[(-1)^(k + 1)*x^(Log[k]*h), {k, 1, Infinity}]
Sum[(-1)^(k + 1)*x^(Log[k]*h), {k, 1, 10000}]
Sum[(-1)^(k + 1)*x^(Round[Log[k]*h]), {k, 1, 10000}]