determine the fourier expansion of the specific function whose definitions in an period is f(t)=t , 0<t<1
Look at your notes for the definition of Fourier series. It should include an
arbitary interval or an interval [0,T] or [-T,T], then renomalise to the interval [0,1].
See also here.
Do what it says and you will have a family of integrals to evaluate, which can be
done using integration by parts.
If you have any more specific questions then please ask
RonL