(*^
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Setup:
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Clear["Global`*"]
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Needs["Algebra`Trigonometry`"]

(* Not Used 
     Needs["Calculus`FourierTransform`"]
*)
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Example 1:
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Clear["Global`*"]
:[font = subsection; inactive; preserveAspect]
Part a) 
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Clear[f];
f[x_]:=x /; x <= N[Pi];
f[x_]:=0 /; x >= N[Pi];
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Plot[f[x],{x,0,2 Pi}];
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c[0]= 1/(2 Pi) Integrate[ x  ,{x,0,1 Pi}]
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Pi/4
;[o]
Pi
--
4
:[font = input; preserveAspect; startGroup]
c[m_]= 1/(2 Pi) Integrate[ x Exp[- I m x] ,{x,0,1 Pi}]
:[font = output; output; inactive; preserveAspect; endGroup]
(-m^(-2) + (E^(-I*m*Pi)*(1 + I*m*Pi))/m^2)/(2*Pi)
;[o]
        -I m Pi
  -2   E        (1 + I m Pi)
-m   + ---------------------
                 2
                m
----------------------------
            2 Pi
:[font = input; preserveAspect]
rule={Exp[+2 I Pi m] ->  1
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     ,Exp[+  I Pi m] -> (-1)^m
     ,Exp[-  I Pi m] -> (-1)^m
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(* Sometimes we have to work out special cases 
 to avoid divide by zero and other problems. *) 
 
term[0]= c[0];
:[font = input; preserveAspect; startGroup]
term[1]= Limit[ c[m] Exp[I m x] + c[-m]  Exp[-I m x] ,m->1]
:[font = output; output; inactive; preserveAspect; endGroup]
(-I/2*E^(-I*x)*(-2*I - 2*I*E^(2*I*x) - Pi + E^(2*I*x)*Pi))/Pi
;[o]
-I  -I x              2 I x         2 I x
-- E     (-2 I - 2 I E      - Pi + E      Pi)
2
---------------------------------------------
                     Pi
:[font = input; preserveAspect; startGroup]
term[m_]= c[m] Exp[I m x] + c[-m]  Exp[-I m x] //.rule //Together
:[font = output; output; inactive; preserveAspect; endGroup]
(E^(-I*m*x)*(-1 + (-1)^m - E^(2*I*m*x) + (-1)^m*E^(2*I*m*x) - I*(-1)^m*m*Pi + I*(-1)^m*E^(2*I*m*x)*m*Pi))/(2*m^2*Pi)
;[o]
 -I m x           m    2 I m x       m  2 I m x         m              m  2 I m x
E       (-1 + (-1)  - E        + (-1)  E        - I (-1)  m Pi + I (-1)  E        m Pi)
---------------------------------------------------------------------------------------
                                           2
                                        2 m  Pi
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series[n_]:= Sum[ term[m] ,{m,0,n}]
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Plot[ Join[{f[x]},Table[series[i],{i,2,5,1}] ] //Evaluate
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:[font = input; preserveAspect]
term2[0] = term[0] //ComplexToTrig //Simplify;
term2[1] = term[1] //ComplexToTrig //Simplify;
:[font = input; preserveAspect; startGroup]
term2[m_]= term[m] //ComplexToTrig //Simplify
:[font = output; output; inactive; preserveAspect; endGroup]
(-Cos[m*x] + (-1)^m*Cos[m*x] - (-1)^m*m*Pi*Sin[m*x])/(m^2*Pi)
;[o]
                m                m
-Cos[m x] + (-1)  Cos[m x] - (-1)  m Pi Sin[m x]
------------------------------------------------
                      2
                     m  Pi
:[font = input; preserveAspect]
series2[n_]:= Sum[ term2[m] ,{m,0,n}]
:[font = input; preserveAspect; startGroup]
series[5]
:[font = output; output; inactive; preserveAspect; endGroup]
Pi/4 - (I/2*E^(-I*x)*(-2*I - 2*I*E^(2*I*x) - Pi + E^(2*I*x)*Pi))/Pi + (E^(-2*I*x)*(-2*I*Pi + 2*I*E^(4*I*x)*Pi))/(8*Pi) + (E^(-3*I*x)*(-2 - 2*E^(6*I*x) + 3*I*Pi - 3*I*E^(6*I*x)*Pi))/(18*Pi) + (E^(-4*I*x)*(-4*I*Pi + 4*I*E^(8*I*x)*Pi))/(32*Pi) + 
 
  (E^(-5*I*x)*(-2 - 2*E^(10*I*x) + 5*I*Pi - 5*I*E^(10*I*x)*Pi))/(50*Pi)
;[o]
     I  -I x              2 I x         2 I x
     - E     (-2 I - 2 I E      - Pi + E      Pi)    -2 I x                 4 I x        -3 I x          6 I x                 6 I x        -4 I x                 8 I x        -5 I x          10 I x                 10 I x
Pi   2                                              E       (-2 I Pi + 2 I E      Pi)   E       (-2 - 2 E      + 3 I Pi - 3 I E      Pi)   E       (-4 I Pi + 4 I E      Pi)   E       (-2 - 2 E       + 5 I Pi - 5 I E       Pi)
-- - -------------------------------------------- + --------------------------------- + ------------------------------------------------ + --------------------------------- + --------------------------------------------------
4                         Pi                                      8 Pi                                       18 Pi                                       32 Pi                                       50 Pi
:[font = input; preserveAspect; startGroup]
series2[5]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Pi/4 - (2*Cos[x])/Pi + Sin[x] - Sin[2*x]/2 + (-2*Cos[3*x] + 3*Pi*Sin[3*x])/(9*Pi) - Sin[4*x]/4 + (-2*Cos[5*x] + 5*Pi*Sin[5*x])/(25*Pi)
;[o]
Pi   2 Cos[x]            Sin[2 x]   -2 Cos[3 x] + 3 Pi Sin[3 x]   Sin[4 x]   -2 Cos[5 x] + 5 Pi Sin[5 x]
-- - -------- + Sin[x] - -------- + --------------------------- - -------- + ---------------------------
4       Pi                  2                  9 Pi                  4                  25 Pi
^*)