d
Potential at origin \(=\left(\mathrm{V}_{1}+\mathrm{V}_{3}+\mathrm{V}_{5}+\ldots \ldots\right)-\left(\mathrm{V}_{2}+\mathrm{V}_{4}+\mathrm{V}_{6}+\ldots .\right)\)

\(\Rightarrow \frac{\mathrm{q}}{4 \pi \varepsilon_{0}}\left[\frac{1}{\mathrm{x}_{0}}-\frac{1}{2 \mathrm{x}_{0}}+\frac{1}{3 \mathrm{x}_{0}} \ldots \infty\right]\)

\(\Rightarrow \frac{q}{4 \pi \varepsilon_{0} x_{0}}\left[1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \ldots . \infty\right]\)

\(\Rightarrow \frac{q}{4 \pi \varepsilon_{0} \mathrm{x}_{0}} \log _{\mathrm{e}}(1+1) \Rightarrow \frac{\mathrm{q}}{4 \pi \varepsilon_{0} \mathrm{x}_{0}} \log _{\mathrm{e}} 2\)