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

Abstract

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