I want to compute the distribution sum of $$\sum_{-\infty}^{\infty}ne^{inx}.$$
Now, this looks awfully familiar to
$$\sum_{n=-N}^{N}e^{inx}=S_{N}[\delta](x)$$
which follows from the fact that, for $\delta\in\mathcal{D}'(\mathbb{T})$, we have $\hat{\delta}(n)=1/2\pi$, so
$$S_{N}[\delta](x)=\sum_{n=-N}^{N}\hat{\delta}(n)e^{inx}.$$
Is there a "standard procedure" for computing these?
I know I have to find $L\in\mathcal{D}'(\mathbb{T})$ such that $$\sum_{n=-N}^{N}ne^{inx}=\sum_{n=-N}^{N}\hat{L}(n)e^{inx}$$