Originally Posted by
ThePerfectHacker Do what topsquark said: $\displaystyle f(x+10) = g(x)$ where $\displaystyle g(x) = x$ on $\displaystyle (-5,5)$. This means, $\displaystyle f(x) = g(x-10)$. Now we expand $\displaystyle g = x$ into a Fourier series on $\displaystyle (-5,5)$. This function is odd so we will just have sine terms. Meaning $\displaystyle \sum_{n=1}^{\infty} a_n \sin \frac{\pi n x}{5}$ where $\displaystyle a_n = \frac{1}{5} \int_{-5}^5 x\sin \frac{\pi n x}{5} dx = \frac{2}{5} \int_0^5 x\sin \frac{\pi n x}{5} dx$ *. Once you find that you will have Fourier for $\displaystyle g(x)$ this means $\displaystyle g(x-10) = \sum_{n=1}^{\infty} a_n \sin \frac{\pi n (x-10)}{5} $.