Find the sum of the series `(1+2x+3x^(2)+(n-1)x^(n-2))` using differentiation.

Explanation

To find the sum of the series \( S = 1 + 2x + 3x^2 + \ldots + (n-1)x^{n-2} \), we can use differentiation. Here’s a step-by-step solution:

### Step 1: Identify the geometric series
We start with the geometric series:
\[ T = 1 + x + x^2 + x^3 + \ldots + x^{n-1} \]
The sum of this series is given by:
...