A charge +q is fixed at each of the points $\overline{X} = \overline{X}_0, \overline{X} = 3\overline{X}_0, \overline{X} = 5\overline{X}_0$ ..... infinite, on the $X-$ axis and a charge -q is fixed at each of the points $x = 2x_0, x = 4x_0, x = 6x_0$ ,..... infinite. Here $x_0$ is a positive constant. Take the electric potential at a point due to a charge $QED$ at a distance r from it to be $Q/(4\pi\epsilon_0 r)$ . Then, the potential at the origin due to the above system of charges is
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$V = \frac{q}{4 \pi \epsilon_0 x_0} \left[ 1 + \frac{1}{3} + \frac{1}{5} + \ldots \right] - \frac{q}{4 \pi \epsilon_0 x_0} \left[ \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \ldots \right]$
$= \frac{q}{4 \pi \varepsilon_0 x_0} \left[ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \right] = \frac{q}{4 \pi \varepsilon_0 x_0} \log_e 2$
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