  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
$k\in \mathbb{N}$.
