Question

Let f (x) = x + x³



for x ∈ [0, π]. What coefficients of the Fourier series of f are zero and why?​

Answer 1

Step-by-step explanation:

Find now the Fourier coefficients for n≠0:

an=1ππ∫−πf(x)cosnxdx=1ππ∫01⋅cosnxdx=1π[(sinnxn)∣∣∣π0]=1πn⋅0=0,

bn=1ππ∫−πf(x)sinnxdx=1ππ∫01⋅sinnxdx=1π[(−cosnxn)∣∣π0]=−1πn⋅(cosnπ−cos0)=1−cosnππn.