Infinite Number of Charges Problem

Let's discuss a problem on Infinite Number of Charges from Physics Olympiad. Try the problem first, and then read the solution here.

The Problem: Infinite Number of Charges

An infinite number of charges, each equal to (q), are placed along the x-axis at (x=1),(x=2),(x=4),(x=8) etc. Find the potential and electric field at the point x=0 due to the set of the charges.

Discussion:

An infinite number of charges, each equal to (q), are placed along the x-axis at (x=1),(x=2),(x=4),(x=8) etc.

Electric potential $$V=\frac{1}{4\pi\epsilon_0}(\frac{q}{1}+\frac{q}{2}+\frac{q}{4}+\frac{q}{8}+...)$$ $$=\frac{1}{4\pi\epsilon_0}(1+\frac{1}{2}+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....)$$
The terms in brackets form a geometric progression of infinite terms whose sum is $$S= \frac{a}{1-r}=\frac{1}{1-1/2}=2$$

Hence the potential $$V=q/2\pi\epsilon_0$$