FourierF[a_, t_] := a.Table[Sin[2 Pi i t], {i, Length[a]}];
FourierAnim[a_, t_] := 
  Module[{A = Accumulate[a*Table[Cos[2 Pi i t], {i, Length[a]}]],
    B = Accumulate[a*Table[Sin[2 Pi i t], {i, Length[a]}]]}
   , PrependTo[A, 0]; PrependTo[B, 0];
   Show[Graphics[
     Table[{Circle[{A[[i]], B[[i]]}, a[[i]]], Darker[Red], 
       If[i != Length@a, 
        Line[{{A[[i]], B[[i]]}, {A[[i + 1]], B[[i + 1]]}}], {Red, 
         Dashed, Line[{{A[[i]], B[[i]]}, {2, B[[i]]}}]}]}, {i, 
       Length@a}], PlotRange -> {{-1.5, 3}, {-1, 1}}], 
    Plot[FourierF[a[[;; -2]], t - \[Tau]], {\[Tau], 2, 3}]]];
a = Table[(1 - (-1)^i)/i, {i, 16}]/Pi;

Manipulate[FourierAnim[a[[;; j]], t], {t, 0, 1}, {j, 1, Length@a, 1}]