I came across a physics question as follows,
A set of positive point charges are kept at the points on the x-axis, such that
And a set of negative point charges are placed on the x-axis, such that,
If the magnitude of the positive point charge is and the magnitude of the negative point charge is , assuming is the absolute permittivity of free space, then prove that the electric potential at the origin is
This is how I proved it.
We know by the definition of electric potential due to a point charge,
where is the separation between the origin and the point where the point charge is placed. Also it is known that the electric potential due to a system of point charges is the algebraic sum of the electric potentials due to the individual point charges.
Hence by applying this to every single point charge given, we get
Factoring out , we get
We know that the infinite summation in the square brackets is the Maclaurin's series expansion of where
Thus we get,
But instead of substituting for the logarithm expansion if we did as I have done in the attached picture, then we get answer as zero. Please tell me where I have gone wrong.