I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier serieses. I read something about convolution, but i wanted to see if there is a trick to do this without using convolution, especially if you don't know know how to do it with convolution :D.
So $ g(x) = x = \sum_{n=1}^{\infty}{2\frac{(-1)^{n+1}}{n}\sin(nx)} $ and $ f(x) = \sin(x)$.
$Fourier[x\sin(x)]= \sum_{n=1}^{\infty}{2\frac{(-1)^{n+1}}{n}\sin(nx)}*\sin(x)$. Using standard trigonometric relationships yields $Fourier[x\sin(x)]= \sum_{n=1}^{\infty}{2\frac{(-1)^{n+1}}{n}\frac{1}{2}*(\cos((n-1)x)-\cos((n+1)x)}$. That almost looks like a fourier series, but how can i turn this into a standard Fourier Series?
Thank you for taking your time and reading my question. Best Regards.