I am trying to compute the coefficients $a_k$ and $b_k$ when k is even then odd. I get a lot of $\frac{C_i(2^{-\Pi})-C_i(1)}{\pi \ln(2)}$ and others that get more and more complicated, with $C_i$ and $\ln$. Since there is no such thing in the answers that are offered, I am clearly wrong.
What am I missing?
Added 20 April 2020.
The problem arises from a different notation for the functions, among others the trigonometric functions. It was written $\sin(2 k+1) \space x$ which meant in fact $\sin((2k+1) \space x)$. Then I obtained immediately the 3rd response. Sorry for that stupid question. But I read what was suggested in the comments about the homework question. The $C_i$ was the cosine integral, which is written Ci(x). Please, can you delete it completely? I really feel like a fool.