Table[
 ListPlot[
  Table[
   it =
    Normal[Series[Log[z + 1] , {z, 0, n}]] /.
     z -> Exp[(\[Pi] - 1/3) I];
   {Re[it], Im[it]},
   {n, 0, a}
   ],
  PlotRange -> {0, 1.8},
  Joined -> True,
  Mesh -> All,
  AxesLabel -> {Style[\[GothicCapitalR], Medium],
    Style[\[GothicCapitalI], Medium]},
  AspectRatio -> 1,
  ImageSize -> 500,
  Epilog -> {it = Log[1 + Exp[(\[Pi] - 1/3) I]];
    Darker[Blue], Text[
     Style[
      HoldForm[
       Subscript[z, n] =
        Sum[(-1)^(k + 1) \[ExponentialE]^(k i (\[Pi] - 1/3))/k, {k,
          1, n}]], 15, Antialiasing -> True
      ], {-0.8, 0.5}
     ],
    Red,
    PointSize[0.01], Point[{Re[it], Im[it]}],
    Text[Style[
      HoldForm[
       Log[1 + \[ExponentialE]^(i (\[Pi] - 1/3))] =
        Sum[(-1)^(k + 1) \[ExponentialE]^(k i (\[Pi] - 1/3))/k, {k,
          1, Infinity}]], 15
      ], {Re[it], Im[it] - 0.2}, {-0.6, 1}]
    }
  ], {a, 5, 77, 3}],
"DisplayDurations" -> PadLeft[{1}, 25, 0.1]
]