(*^
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Example 1:
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Pick some Fourier coefficients:
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c={5,0,2,0,3};
BarChart[c,PlotRange->{{0, 10},All}];
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Make a Fourier Series:
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Clear[f];
f[x_]= Sum[ c[[n]] Sin[n x] ,{n,1,5}]
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5*Sin[x] + 2*Sin[3*x] + 3*Sin[5*x]
;[o]
5 Sin[x] + 2 Sin[3 x] + 3 Sin[5 x]
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Plot the Fourier Series:
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data= Table[f[x] //N,{x,0,2 Pi,2 Pi/2^7}];
ListPlot[data ,PlotJoined->True];
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Note, I Dropped the c_0 element so that the 
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Compare this with first plot.

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Same thing with ListPlot instead. 
(ListPlot is more general for our purposes.)
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MathPictureEnd

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Problem 1: A Gaussian:
:[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect; endGroup]
Clear[f];
f[x_]= E^(-x^2);
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Problem 2: A Step Function:
:[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect]
Clear[f];
f[x_]:= 1 /; x<=N[Pi];
f[x_]:= 0 /; x>=N[Pi];
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ifdata= InverseFourier[ Chop[fdata,0.5] ];
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Example 2: Beats:
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Play[  Sin[2 Pi 440 t]+ Sin[2 Pi 444 t],{t,0,1}]
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pop
% End of sampled sound
% End of Graphics
MathPictureEnd

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Sound["<<>>"]
;[o]
-Sound-
:[font = section; inactive; noKeepOnOnePage; preserveAspect; startGroup]
Problem 3: Trumpet vs. Clarinet
:[font = input; dontNoPageBreakInGroup; noKeepOnOnePage; preserveAspect]
trpt= Sum[ 1/n Sin[2 Pi 220 n t] ,{n,1,5,1}]
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clair= Sum[ 1/n Sin[2 Pi 220 n t] ,{n,1,5,2}]
^*)