I have $b_k=\frac{1}{k(ln[1+k])^2}$ for all $k\in\mathbb{N}$. We have the triginometric series: $$\sum_{k=1}^{\infty}b_k\sin(kx),$$ and has to show that the sum is odd, and write on exponential form. That the sum is odd is because $\sin(kx)$ is odd, and the sum of odd products must be odd
However, I have a hard time writing it on exponential form, but have come to the following, but I don't know if it is correct: $$b_k=i(f_k-f_{-k})$$ $$\frac{b_k}{i}=f_k-f_{-k}$$ We can see that $a_k=0$ from the sum above, so $$0=a_k=f_k+f_{-k}\quad\quad\Rightarrow\quad\quad f_k=-f_{-k}$$ So I have: $$\frac{b_k}{i}=2f_k,$$ which gives: $$f_k=\frac{b_k}{2i}$$ so the exponential form becomes: $$\sum_{k=0}^\infty f_ke^{ikx}$$
I don't know how to verify whether it is correct.