It looks like you've uploaded a mathematical problem involving series. Let's break it down and address the different parts:

Problem Breakdown:

Part (a)

You need to find the sum of the series:
$\sum_{n=1}^{\infty} n x^{n-1}, \quad |x| < 1.$

Part (b)

For this part, we need to find the sum of the following series:

1. $\sum_{n=1}^{\infty} n x^n, \quad |x| < 1$
2. $\sum_{n=1}^{\infty} \frac{n}{9^n}$

Part (c)

For part (c), we have three series to find the sums of:

1. $\sum_{n=2}^{\infty} n(n-1) x^n, \quad |x| < 1$
2. $\sum_{n=2}^{\infty} \frac{n^2 - n}{2^n}$
3. $\sum_{n=2}^{\infty} \frac{n^2}{2^n}$

Would you like me to explain how to solve each part step by step? Or would you like the solutions for any specific part first? Let me know!