The sum of the infinite series 1² + 2² x + 3² x² + 4² x³ + ..... (x < 1) is _______.

Correct option is A

Given Series:

​$$1^2 + 2^2x + 3^2x^2 + 4^2x^3 + \dots \quad \text{where } x < 1.$$​​

This is an infinite series where the general term is:

​$$T_n = n^2 x^{n-1}$$​​

We aim to find the sum of this series.

Formula Used:

For an infinite series of the form:

​$$\sum_{n=1}^\infty n^2 x^{n-1}$$​

the sum is given by:

S = $$\frac{1+x}{(1-x)^3}, \quad \text{valid for } |x| < 1$$​

Solution:

1. The series follows the pattern $$1^2 + 2^2x + 3^2x^2 + 4^2x^3 + \dots$$​ 

2. Applying the formula directly for the sum:

​$$S = \frac{1+x}{(1-x)^3}$$​

Answer:

​$$\mathbf{A. \, \frac{1+x}{(1-x)^3}}$$​