I am trying to evaluate Fourier series expansion of flow rate as a function of crank angle in a duplex positive displacement pump. I have evaluated Fourier series coefficients for the function. However the Fourier series expansion does not match the parent function even if I sum the series to a large number of terms. I would appreciate any help in identifying what I could be doing wrong.

Clear[fsum, fourier, n, nmax, coeff];

f = 5; n =.; t =.; \[Omega] = 2 \[Pi] f; r = 1.; L = 
 5 r; \[Theta] = \[Omega] t;

Clear[P1, P2, coeff, a, b];
P1[\[Theta]_] := -r Sin[\[Theta]] - (r^2 Sin[\[Theta]] Cos[\[Theta]])/
    Sqrt[L^2 - r^2 (Sin[\[Theta]])^2];
P2[\[Theta]_] := P1[\[Theta] + Pi];

fsum[\[Theta]_] := Max[P1[\[Theta]], 0] + Max[P2[\[Theta]], 0];

coeff[n_Integer] := 
  coeff[n] = (1/\[Pi]) Quiet@
     NIntegrate[
      fsum[\[Theta]] Exp[-I  n \[Theta]], {\[Theta], -\[Pi], -3/
         4 \[Pi], -\[Pi]/2, -1/4 \[Pi], 0, 1/4 \[Pi], 1/2 \[Pi], 
       3/4 \[Pi], \[Pi]}, AccuracyGoal -> 8];

a[n_] = Re@coeff[n];
b[n_] = Im@coeff[n];

Clear[fourier, ul];
ul = 1000;

fourier[n_] := 
  a[0] /2 + 
   Sum[Re@coeff[n] Cos[n \[Theta]] + Im@coeff[n] Sin[n \[Theta]], {n, 
     1, ul}];

Plot[{fsum[\[Theta]], fourier[n]}, {\[Theta], 0, 2  \[Pi]}, 
 Frame -> True, Axes -> True, PlotRange -> { {0, 2 \[Pi]}, {0, 1.5}}, 
 FrameLabel -> {"Crank Angle (rad)", "Flow rate"}, 
 GridLines -> Automatic, AspectRatio -> .65, 
 PlotStyle -> {{Red, Thickness[0.005]}, {Green, 
    Thickness[0.0051]}, {Gray, Thickness[0.01]}}, 
 BaseStyle -> {FontWeight -> "Bold", FontSize -> 15, 
   FontFamily -> "Arial"}]

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