$\text{Find the Fouruer series of $f(x)=x\sin(x)$ on $[-\pi,\pi]$ and determine its sum for all |x| $\leq$ $\pi$}$
I found the fourier series. That wasn't the hard part. I got:
$$1+2\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2-1}\cos(nx)$$
Which I am fairly confident in.
I'm confused as to how to find the sum of the fourier series though.
I thought I would just take the average of the right and left endpoints and that would be my sum but I am not sure...
Could someone tell me how I would go about finding the sum?
Extend $f$ to $f^*$: a periodic function of period $2\pi$ which equals $f$ on $[-\pi,\pi]$. Then $f^*$ is continuous and so its Fourier series converges pointwise to $f^*$. In particular, the sum of your series is $x \sin x$ for all $x \in[-\pi,\pi]$.