I'd really love your help with this one: I got this Fourier series for $f(x)=x$ in $[-\pi,\pi]$: $\sum_{1}^{\infty}\frac{2(-1)^{n+1}\sin(nx)}{n}$ and I need to check if it's (i) pointwise converges,(ii) uniformly converges and if (iii) $\sum_{1}^{\infty}\left|\frac{2(-1)^{n+1}\sin(nx)}{n}\right|$ converges.. I tried to do it, and I have couple of specific questions that interrupt me.
(i) Can I claim that the series is pointwise due to Leibniz test? since $\frac{\sin(nx)}{n} \to 0$? I was told that I need to use Lipschitz continuity here. How do I use it?
(ii) I was told that this series is not uniformly converges since $f(x)$ is not continuous in this part. How come? Is it because we always see those functions who we compute their Fourier series as periodic functions with period of $[0,2\pi]$ so in our case the function in $0$ is not continues?
(iii) How come I can't use Dirichlet test for series and claim that the $\sum_{1}^{\infty}\left|\frac{2(-1)^{n+1}\sin(nx)}n\right|$ does converges? The test requires a multiple of a series that monotonic converges to $0$ and $\sum_{1}^{N}|\sin(nx)|
Thanks you so much guys!