`1^(2)+ 3^(2)x+ 5^(2)x^(2)+ 7^(2)x^(3)+.....`

Answer

To solve the given series \(1^2 + 3^2 x + 5^2 x^2 + 7^2 x^3 + \ldots\), we can express it in a more manageable form.

### Step-by-Step Solution:

1. **Identify the Pattern**:
The series consists of squares of odd numbers multiplied by increasing powers of \(x\). The general term can be expressed as:
\[ a_n = (2n - 1)^2 x^{n-1} \]
where \(n\) starts from 1.

2. **Write the Series**:
The series can be rewritten as:
\[ S = \sum_{n=1}^{\infty} (2n - 1)^2 x^{n-1} \]

3. **Use the Formula for the Sum of the Series**:
To find the sum of this series, we can use the known formula for the sum of squares of odd numbers. The sum of the series can be derived using generating functions or known results. The sum of the series can be expressed as:
\[ S = \frac{1}{(1 - x)^3} \]
for \(|x| < 1\).

4. **Final Result**:
Therefore, the value of the series is:
\[ S = \frac{1}{(1 - x)^3} \]