piecewise

Clear[x, y, a, b, c, d, e]

s[x_] := N[\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(1\), \(m\)]\(
\*FractionBox[\(1\), \(2  y\)] Sign[Sin[
\*FractionBox[\(\[Pi]\ x\), \(y\)]]] \[DifferentialD]y\)\)]
a = s[50]; b = s[100]; c = s[150]; d = s[200]; e = s[250]; f = s[300];

k = 50; v1 = Quiet[(NIntegrate[-(1/2) (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)), {x, 0, k}])];
k = 50; v2a = Quiet[(NIntegrate[1/2 (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)) - N[1/2 (Log[50/50] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(50\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\))] + a, {x, 0, k}])];
k = 100; v2b = Quiet[(NIntegrate[1/2 (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)) - N[1/2 (Log[50/50] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(50\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\))] + a, {x, 0, k}])];

y = 150; m = 50;
data = Accumulate[
   Total[Take[Flatten[ConstantArray[#, Ceiling[(y)/Length@#]]], y] & /@
       Table[Join[ConstantArray[1/row, row],
        ConstantArray[0, row]], {row, 1, m}]] - Sum[1/k, {k, 1, m}]/2];
Show[ListPlot[data, Frame -> True],
 ListLinePlot[Table[Quiet[NIntegrate[-(1/2) (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)), {x, 0, k}]], {k, 0, 50}],
  DataRange -> {0, 50}, InterpolationOrder -> 1,
  PlotStyle -> {Red, Thick}],
 ListLinePlot[Table[Quiet[NIntegrate[1/2 (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)) - N[1/2 (Log[50/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(50\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\))] + a, {x, 0, k}]], {k, 50,
     100}] + v1 - v2a, DataRange -> {50, 100},
  InterpolationOrder -> 1, PlotStyle -> {Red, Thick}],
 ListLinePlot[Table[Quiet[NIntegrate[-(1/2) (Log[x/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(x\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\)) + N[1/2 (Log[100/m] + \!\(
\*SubsuperscriptBox[\(\[Sum]\), \(n = 2\), \(100\)]\(
\*SuperscriptBox[\((\(-1\))\), \(n\)] Log[
\*FractionBox[\(n\), \(n - 1\)]]\)\))] + b, {x, 0, k}]], {k, 100,
     150}] + 2 v1 + 2 v2b - b, DataRange -> {100, 150},
  InterpolationOrder -> 1, PlotStyle -> {Red, Thick}]]