You examples are easy, I was hoping for harder ones ;) This is from the definition.

    f3[x_] := Piecewise[{{1 - x^2 , x < 0}, {1 + x^2, x > 0}}];
    FourierSeries[f3[x], x, 3]

![Mathematica graphics](http://i.stack.imgur.com/JPS7B.png)


A quick Manipulate:

![enter image description here][1]

    Manipulate[
     r = FourierSeries[f[x], x, n];
     Show[Plot[r, {x, -2 Pi, 2 Pi}, Frame -> True], Plot[f[x], {x, -2 Pi, 2 Pi}, 
          PlotStyle -> {Thick, Red}]],
     Grid[{
       {Control[{{n, 3, "how many terms?"}, 1, 20, 1}], Dynamic[n]}
       }],
     ContinuousAction -> False,
     SynchronousUpdating -> True,
     Initialization :>
      (
       f[x_] := Piecewise[{{1 - x^2 , x < 0}, {1 + x^2, x > 0}}]
       )
     ]

![Mathematica graphics](http://i.stack.imgur.com/rTu0p.png)

And if you meant them to be different functions:

    f1[x_] := Piecewise[{{1 - x^2 , x < 0}, {0, True}}];
    f2[x_] := Piecewise[{{1 + x^2 , x > 0}, {0, True}}];
    FourierSeries[f1[x], x, 3]

![Mathematica graphics](http://i.stack.imgur.com/8FfCb.png)

    FourierSeries[f2[x], x, 3]

![Mathematica graphics](http://i.stack.imgur.com/wQM2d.png)


You can use the definition of the $c_k$ also by using `FourierParameters` to make it match the textbook you are using. So make sure to look at `FourierParameters` and adjust it as needed else you'll get different looking result from the textbook if the textbook does not use the default setting used by Mathematica.


  [1]: http://i.stack.imgur.com/GyGAr.gif