A charge $$+q$$  fixed at each of the points $$x = {x_0},x = 3{x_0},x = 5{x_0},...$$      upto $$\infty $$ on $$X$$-axis and charge $$-q$$  is fixed on each of the points $$x = 2{x_0},x = 4{x_0},...$$    upto $$\infty .$$ Here $${x_0}$$ is a positive constant. Take the potential at a point due to a charge $$Q$$ at a distance $$r$$ form it to be $$\frac{Q}{{4\pi {\varepsilon _0}r}},$$  then the potential at the origin due to above system of charges will be:

Solution :
$$\eqalign{ & V = \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{q}{{{x_0}}} + \frac{q}{{3{x_0}}} + \frac{q}{{5{x_0}}} + ...} \right] + \frac{1}{{4\pi {\varepsilon _0}}}\left[ {\frac{{\left( { - q} \right)}}{{2{x_0}}} + \frac{{\left( { - q} \right)}}{{4{x_0}}} + \frac{{\left( { - q} \right)}}{{6{x_0}}} + ...} \right] \cr & = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{x_0}}}\left[ {1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + ....} \right]a \cr & = \frac{1}{{4\pi {\varepsilon _0}}}\frac{q}{{{x_0}}}{\log _e}\left( {1 + 1} \right) \cr & = \frac{{q{{\log }_e}2}}{{4\pi {\varepsilon _0}{x_0}}} \cr} $$