Consider a function $f: [0, L] \to \Bbb R$ and $k$ natural number. Suppose that $f', f'', ...$ $f^{n-1}$ are continuous and that $f^{k}$ is absolutely integrable. Show
$$ \left| \int_{0}^{L}{f(x)\sin \left(\frac{nx\pi}{L}\right) dx}\right| \leq \dfrac{C}{n^{k}}$$
$n=1,2,3...$, where $C$ is a positive constant.
Progress
I tried integration by parts: $$\int_{0}^{L}{f(x)\sin\left(\dfrac{nx\pi}{L}\right)dx} = \dfrac{-L}{n\pi}(\cos(n\pi)f(L)-f(0))+\dots $$ And $n^{k}$ is obtained, but my question is whether this well bounded so, because it could continue to take many $n$'s of the integral.
