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 $$e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots =\mathrm{Pf}\frac{e^{it}}{(1+e^{it}{)}^2}+i\pi \sum _{n\in \mathbb{Z}}{\delta }_{(2n+1)\pi }^{\prime },$$ where $\mathrm{Pf}\frac{e^{it}}{(1+e^{it}{)}^2}\in {\mathcal{D}}^{\prime }(\mathbb{R})$ is the $2\pi$-periodic distribution given by \begin{eqnarray*} \left\langle {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} ,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*} $\phi \in \mathcal{D}(\mathbb{R})$ with support $\text{supp}(\phi )\subset (-\delta ,2\pi +\delta )$, 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 $k\in \mathbb{N}$.

31 pages, 5 figures (Typos corrected.)