I have this set of Fourier coefficients that I want to sum together. However, this what I have written is wrong in its last line.
I cannot figure out what is wrong with the last line, since Mathematica solved this.
\begin{equation} \begin{split} &\alpha_k^j=\frac{\eta_{j}(-1)^k}{\pi k}\sin\frac{\pi k }{n}\cos(2j-1)\frac{\pi k }{n}{}\\ &\alpha_k^1=\frac{2(-1)^k}{\pi k}\sin\frac{\pi k }{4}\cos\frac{\pi k }{4}{}\\ &\alpha_k^2=\frac{(-1)^k}{\pi k}\sin\frac{\pi k }{4}\cos\frac{3\pi k }{4}{}\\ &\alpha_k^3=\frac{2(-1)^k}{\pi k}\sin\frac{\pi k }{4}\cos\frac{5\pi k }{4}{}\\ &\alpha_k^4=\frac{3(-1)^k}{\pi k}\sin\frac{\pi k }{4}\cos\frac{7\pi k }{4}{}\\ &\alpha_k=\sum_{j=1}^4\alpha_k^j=\frac{(-1)^k}{2\pi k}\bigg(\sin\frac{k\pi}{2}-\sin\pi k + 3\sin2\pi k- \sin\frac{3k\pi}{2}\bigg) \end{split} \end{equation}
Does anyone have a hint here?
Thanks