Question

Derive fourier series for constant function

Answer 1

f(x)≈A02+∑∞1Ancos(2πLnx)+Bnsin(2πLnx)f(x)≈A02+∑1∞Ancos⁡(2πLnx)+Bnsin⁡(2πLnx) 

If f(x)f(x) is equal to a constant; f(x)=cf(x)=c, it should be clear that the sines and cosines must have no contribution to the Fourier series. Their coefficients An,Bn,if n>0An,Bn,if n>0 are zero.
Then the Fourier series reduces to f(x)≈A02f(x)≈A02, a constant.