A summation method based on the Fourier series of periodic distributions and an example

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that $$e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} +i\pi \displaystyle \sum_{n\in \mathbb{Z}} \delta'_{(2n+1)\pi},$$ where ${\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}}\in \mathcal{D}'(\mathbb{R})$ is the $2\pi$-periodic distribution given by \begin{eqnarray*} \left\langle {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} ,\varphi \right\rangle &=& \lim_{\epsilon\searrow 0} \left( \int_{(-\delta,\pi-\epsilon)\cup(\pi+\epsilon,2\pi+\delta)}\frac{\varphi(t) e^{it}}{(1+e^{it})^2}dt -\frac{\varphi(\pi)}{\tan (\epsilon/2)}\right) \end{eqnarray*} $\varphi \in \mathcal{D}(\mathbb{R})$ with support $\textrm{supp}(\varphi)\subset (-\delta,2\pi+\delta)$, where $\delta\in (0,\pi)$. Applying the generalised summation method, we determine the sum of the divergent series $1+2+3+\cdots$, and more generally $1^k+2^k+3^k+\cdots$ for $k\in \mathbb{N}$.