```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}}]
)
]

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
```