A conducting sphere `S_1` of radius `r` is attached to an insulating handle. Another conduction sphere `S_2` of radius `R` is mounted on an insulating stand. `S_2` is initially uncharged. `S_1` is given a charge `Q` brought into contact with `S_2` and removed. `S_1` is recharge such that the charge on it is again `Q` and it is again brought into contact with `S_2` and removed. This procedure is repeated `n` times.
a. Find the electrostatic energy of `S_2` after `n` such contacts with `S_1`.
b. What is the limiting value of this energy as `nrarroo` ?

Explanation

To solve the problem step by step, we will analyze the situation involving two conducting spheres, \( S_1 \) and \( S_2 \), and determine the electrostatic energy of \( S_2 \) after \( n \) contacts with \( S_1 \), as well as the limiting value of this energy as \( n \) approaches infinity.

### Step 1: Understanding the Charge Transfer Process
When the charged sphere \( S_1 \) (with charge \( Q \)) comes into contact with the uncharged sphere \( S_2 \), charge will redistribute between the two spheres until they reach the same potential.

1. **Initial Charges**:
- Charge on \( S_1 \): \( Q \)
- Charge on \( S_2 \): \( 0 \)
...