How can I expand the Fourier coefficient The Fourier series is

How can I expand the Fourier coefficient $a_n$ and $b_n$ to the n order? $x\in [-Pi,Pi]$,$n \gt 0$.

Sadly there isn't a FourierTrigCoefficient built in Mathematica up to now, but you can use my easyFourierTrigSeries for the task.
Let me illustrate the solving process with 
$$f(x)=e^{-x}$$
If we need to find the general term of $a_n$ and $b_n$ for $x\in(-1,1)$, just write
Clear@f
f[x_] = Exp[-x];
series = easyFourierTrigSeries[f[x], {x, -1, 1}]


The formula of $a_n$ and $b_n$ is already clear at this stage, simply copy them manually from the output isn't a bad idea, still, we can extract them progammatically in the following way:
Cases[series, a_ Cos[_] + b_ Sin[_] :> {a, b}, Infinity]


