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}]
