Linear chain

The electrostatic energy is a sum over all ions in the chain at positions $x_j = j \cdot d$ and charge $q_j = (-1)^j \cdot e$ : $\begin{aligned} W &= \frac{1}{4 \pi \varepsilon_0} \sum_{j \not= 0} \frac{q_0 q_j}{|x_0 - x_j|} = \frac{e^2}{4 \pi \varepsilon_0 d} \sum_{j \not= 0} \frac{(-1)^j}{|j|} \\ &= -\frac{e^2}{4 \pi \varepsilon_0 d} \cdot 2 \sum_{j = 1}^\infty \frac{(-1)^{j+1}}{j} = -\frac{e^2 \ln 2}{2 \pi \varepsilon_0 d} \\ &= 3.198 \cdot 10^{-18} \,\text{J} = 19.96 \,\text{eV} \approx 20 \,\text{eV} \end{aligned}$ With the help of the logarithm series $\log(1 + x) = \sum_{j =1}^\infty \frac{(-1)^j}{j} \cdot x^j$ .