Needs["Graphics`Graphics`"]

Example 1:

Pick some Fourier coefficients:

c={5,0,2,0,3};
BarChart[c,PlotRange->{{0, 10},All}];

Make a Fourier Series:

Clear[f];
f[x_]= Sum[ c[[n]] Sin[n x] ,{n,1,5}]

Plot the Fourier Series:

data= Table[f[x] //N,{x,0,2 Pi,2 Pi/2^7}];
ListPlot[data ,PlotJoined->True];

Fourier Transform Series:

fdata=Fourier[data];

(* 

Note, I Dropped the c_0 element so that the 
numbers match up better; i.e., c_1 is at 1, ... 
Compare this with first plot.

*)

BarChart[fdata //Drop[#,1]& //Abs 
     ,PlotRange->{{0, 10},All}
     ];

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

ListPlot[fdata //Drop[#,1]& //Abs 
     ,PlotJoined->True
     ,PlotRange->{{0,20},All}
     ,AxesOrigin->{0,0}
     ];

Problem 1: A Gaussian:

Clear[f];
f[x_]= E^(-x^2);

Problem 2: A Step Function:

Clear[f];
f[x_]:= 1 /; x<=N[Pi];
f[x_]:= 0 /; x>=N[Pi];

ifdata= InverseFourier[ Chop[fdata,0.5] ];

Example 2: Beats:

Play[  Sin[2 Pi 440 t]+ Sin[2 Pi 444 t],{t,0,1}]

Problem 3: Trumpet vs. Clarinet

trpt= Sum[ 1/n Sin[2 Pi 220 n t] ,{n,1,5,1}]

clair= Sum[ 1/n Sin[2 Pi 220 n t] ,{n,1,5,2}]