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 &=& \sum_{n\in \mathbb{Z}} \int_0^{2\pi} \frac{(t-\pi)^2 e^{it}}{(1+e^{it})^2} \int_0^1 (1-\theta) \varphi''((2n+1)\pi +\theta (t-\pi)) d\theta dt, \end{eqnarray*} for $ \varphi \in \mathcal{D}(\mathbb{R})$. 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 $k\in \mathbb{N}$.

中文翻译：

---

基于周期性分布傅里叶级数的求和方法和卡西米尔效应中的一个例子

考虑基于周期性分布的傅立叶级数的广义求和方法。表明 $$ e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^ {it}}{(1+e^{it})^2}} +i\pi \displaystyle \sum_{n\in \mathbb{Z}} \delta'_{(2n+1)\pi}, $$ where ${\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}}\in \mathcal{D}'(\mathbb{ R})$ 是 $2\pi$-周期分布由 \begin{eqnarray*} \left\langle {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e ^{it})^2}} ,\varphi \right\rangle &=& \sum_{n\in \mathbb{Z}} \int_0^{2\pi} \frac{(t-\pi)^2 e^{it}}{(1+e^{it})^2} \int_0^1 (1-\theta) \varphi''((2n+1)\pi +\theta (t-\pi) ) d\theta dt, \end{eqnarray*} 为 $ \varphi \in \mathcal{D}(\mathbb{R})$。应用广义求和法，