Find a function such that $ \int_{-\pi}^{\pi} f(x)\sin(nx)dx = \frac{(-1)^n}{\sqrt n} $ and $ \int_{-\pi}^{\pi} f(x)\cos(nx)dx = 0 $ As the title states, I must say if the function exists or not.
I'm not sure where to begin...
Is there a general method or approach to finding this type of functions?
All I can think is that $ (-1)^n = \cos(\pi n) $ but I don't know how the $ \sqrt n $ could appear.

The function $$\frac{i}{2} \left\{\textrm{Li}_{1/2}[-e^{-ix}] - \textrm{Li}_{1/2}[-e^{ix}] \right\}= \sum_{k=1}^\infty (-1)^k \left[ \frac{e^{-ikx}}{k^{1/2}}- \frac{e^{ikx}}{k^{1/2}}\right]=\sum_{k=1}^\infty (-1)^{k} \frac{\sin kx }{k^{1/2}} $$ satisfies the requirements.
Here $\textrm{Li}_{\alpha}(x)$ is the polylogarithm function.
